Is there a specific reason that most elves aren't very fond of the dwarves in Tolkien's writings?
Overall Elves and Dwarves were mutually respectful, if not friendly, in the First Age:
Ever cool was the friendship between the Naugrim and the Eldar, though much profit they had one of the other; but at that time those griefs that lay between them had not yet come to pass, and King Thingol welcomed them. But the Naugrim gave their friendship more readily to the Noldor in after days than to any others of Elves and Men.
They collaborated in works like the creation of the fortress of Menegroth in Doriath:
the Naugrim laboured long and gladly for Thingol, and devised for him mansions after the fashion of their people, delved deep in the earth. Where the Esgalduin flowed down, and parted Neldoreth from Region, there rose in the midst of the forest a rocky hill, and the river ran at its feet. There they made the gates of the hall of Thingol, and they built a bridge of stone over the river, by which alone the gates could be entered. Beyond the gates wide passages ran down to high halls and chambers far below that were hewn in the living stone, so many and so great that that dwelling was named Menegroth, the Thousand Caves.
Thingol's murder was the trigger that changed the relationship - Thingol had thought to combine the greatest work of the Elves (a Silmaril) and that of the Dwarves, the Nauglamir. He hired dwarves from Nogrod to do so, who seized the Silmaril and refused to return it. When Thingol attempted to recover it, the dwarves slew him and were then slain in turn:
Thingol went down alone to their deep smithies, and sat ever among them as they worked. In time his desire was achieved, and the greatest of the works of Elves and Dwarves were brought together and made one; and its beauty was very great, for now the countless jewels of the Nauglamír did reflect and cast abroad in marvellous hues the light of the Silmaril amidmost. Then Thingol, being alone among them, made to take it up and clasp it about his neck; but the Dwarves in that moment withheld it from him, and demanded that he yield it up to them. ... they rose up about him, and laid hands on him, and slew him as he stood.
Then the Dwarves taking the Nauglamír passed out of Menegroth and fled eastwards through Region. But tidings went swiftly through the forest, and few of that company came over Aros, for they were pursued to the death as they sought the eastward road. ... Yet two there were of the slayers of Thingol who escaped from the pursuit on the eastern marches, and returned at last to their city far off in the Blue Mountains; and there in Nogrod they told somewhat of all that had befallen, saying that the Dwarves were slain in Doriath by command of the Elvenking, who thus would cheat them of their reward.
The two dwarven escapees then incited their people to war over this - the result was the Sack of Doriath, and the destruction of one of the great Elven realms:
the Dwarves held on their way, and passed over the great bridge, and entered into Menegroth; and there befell a thing most grievous among the sorrowful deeds of the Elder Days. For there was battle in the Thousand Caves, and many Elves and Dwarves were slain; and it has not been forgotten. But the Dwarves were victorious, and the halls of Thingol were ransacked and plundered.
After that there was a level of mutual distrust between the races, with occasional thawing of the relationship - Eregion and Moria being another example, as seen in the carving on the doors to the Mines of Moria.
All quotes from the Silmarillion.
The animosity between elves and dwarves goes back farther than any of the events mentioned in the other answers, it goes back right to the creation of the dwarves.
From "Of Aulë and Yavanna" (Chapter 2 of "Quenta Silmarillion"):
The dwarves were created by Aulë because he was impatient for the coming of the Children of Ilúvatar. Aulë wanted:
to have learners to whom he could teach his lore and his crafts, ...
Ilúvatar knew of Aulë's creation and he came to Aulë and chastised him, pointing out that Aulë couldn't create life, only he could do that. But despite Aulë's overstepping, Ilúvatar gave life to the dwarves, but required that they sleep until after the elves had awoken and he warned Aulë:
"But when the time comes I will awaken them, and they shall be to thee as children; and often strife shall arise between thine and mine, the children of my adoption and the children of my choice."
So basically, the elves and dwarves were rival creations, destined to be at odds from the very beginning.
There's a very long enmity between them. It dates right back to the First Age, when Dwarves murdered the Elf King Thingol of Doriath after a quarrel over the Nauglamir, the Necklace of the Dwarves, which had been set with the last remaining Silmaril, an Elf-made jewel. You can read the full story in The Silmarillion, and in even more details in The Children of Hurin.
Later there was a bit of a rapprochement, especially between the Dwarves of Moria and the Elves of neighbouring Hollin. But the Elves blamed the Dwarves for awakening the Balrog under Moria, and were never friendly again until the events of the Lord of the Rings.
The History--and ill will--is spoken quite clearly in the Silmarillion; however, I give Peter Jackson props for his little spin on things. Since he has limited time and many movie goers will not be familiar with the "old" histories (many find the Silmarillion TOO much like a history book) he needed to create the hostility between Thranduil and Thorin. What better way than to have Thorin see the elves watching their destruction and turning their backs! Those of us familiar with Tolkien's other works know the whole truth, but for those just enjoying an AWESOME movie?...nicely done Mr. Jackson!
While Illuvatar did declare there'd be eternal strife between the children of his adoption and the children of his choice in the very beginning, fating it to be so, as the other answers mention, and the dwarves did kill King Thingol, the earthly origin of their mutual dislike is a little more prosaic.
Elves thought dwarves were ugly. The Elven name for the dwarves is "naugrim", meaning "stunted people" - not very flattering. Even their language was, to the Elves "cumbrous and unlovely." (Silmarillion) Friendship between the Dwarves and Elves was always "very cool" for this reason. Even the Noldor, the Elves the Dwarves got along best, didn't do much for them: "for the dwarves were secret and quick to resentment, and Caranthir was haughty and scarce concealed his scorn for the unloveliness of the Naugrim, and his people followed their lord." (Of the Return of the Noldor) They simply had very different values and interests: the Dwarves gave most of their reverence to Aule, and hated and feared the sea; most Elves were much more broad in their reverence and loved the sea.
The first meeting of Dwarves and Elves was in the First Age. The first Dwarves the Elves met were the Petty Dwarves, exiles from the dwarven kingdoms. And how did the Elves treat these Dwarves? Well, by hunting them like animals - because they didn't even recognize they were beings with souls.
From the War of the Jewels:
The Eldar did not at first recognize these as Incarnates, for they seldom caught sight of them in clear light. They only became aware of their existence indeed when they attacked the Eldar by stealth at night, or if they caught them alone in wild places. The Eldar therefore thought that they were a kind of cunning two-legged animals living in caves, and they called them Levain tad-dail, or simply Tad-dail, and they hunted them. But after the Eldar had made the acquaintance of the Naugrim, the Tad-dail were recognized as a variety of Dwarves and were left alone. There were then few of them surviving, and they were very wary, and too fearful to attack any Elf, unless their hiding-places were approached too nearly.
Now, the other Dwarves did despise the Petty Dwarves - but:
they still acknowledged their kinship and resented any injuries done to them. Indeed it was one of their grievances against the Eldar that they had hunted and slain their lesser kin, who had settled in Beleriand before the Elves came there. This grievance was set aside, when treaties were made between the Dwarves and the Sindar, in consideration of the plea that the Petty-dwarves had never declared themselves to the Eldar, nor presented any claims to land or habitations, but had at once attacked the newcomers in darkness and ambush. But the grievance still smouldered, as was later seen in the case of Mîm, the only Petty-dwarf who played a memorable part in the Annals of Beleriand.
The Ruin of Doriath can only be understood in this context. Elves and Dwarves never had much trust or much liking for each other. Their relationship began with genocide. They stood together insofar as they both (well, mostly) opposed Morgoth. This can also help us why even the Wise would blame the Dwarves for stirring up the Balrog in Moria (again telling that the Elvish name for the greatest of the Dwarven cities is "black pit") even though it's irrational: Elves are predisposed by nature and thousands of years of history to be prejudiced against dwarves.
Actually, Moria is a great point. Even the Elves friendly with the Dwarves graffiti-ed the greatest of the Dwarven kingdoms with the words "black chasm" - shows you what they thought of it. ;)
When Hurin was released from Morgoth's seat, Morgoth had planted in his mind that Thingol and Melian of Doriath had sent his children to their death and his wife to a life of wandering. He went to the deserted Elven kingdom of Nargothrond, which had been overthrown by a dragon when his son, Turin, was there. He slayed Mim the petty-dwarf who had taken up abode there but had also been an earlier enemy of his son. Hurin stayed there several days and then left taking one item with him.
He then went to Doriath and came before Thingol and Melian, handing over to them a necklace made for a dead Elven king by the dwarves, in order to pay them for taking "care" of his wife and children.
Melian released him from the lies of Morgoth and he realised that they had only ever helped his family. He still gave the necklace of the dwarves to the Elf king.
Thingol had obtained one of the three silmarils from Beren as a dowery for Luthien. It had ever grown on his mind and he wanted to be with it often (much like the rings later on in Middle-Earth history). After examining the great workmanship of the dwarven necklace he went to the refugee dwarves who worked as smithies in his kingdom. He told them that he wanted them to fasten the silmaril into the necklace so that he could wear it.
The dwarves did as instructed but lusted for the silmaril and the necklace. Thingol came to watch their work ever often. He was there when the necklace was completed and went to take it from them. The dwarves would NOT give it to the king saying that it had come from a robber of men and the original necklace HAD NOT BEEN CREATED FOR THINGOL.
Thingol got mad and insulted the dwarves, up to calling them "stunted people." The dwarves grew angry and slew the king. As they made their escape WITH THE NECKLACE, elven warriors came upon them and slew several of their numbers. The rest fled to their people and told their cheif that THINGOL originally had attacked them. An army of dwarves was gathered and sent to slaughter the elves of Doriath.
The reasons may also be differences in their attitude, worldview and such things. First:
dwarves are very warlike race while elves prefer peaceful life (though when pressed by dangers they become great warriors and sometimes distrustful of strangers often they isolate themeselves, like many Wood Elves, Noldor can be more aggressive and overall elven individuals can show intolerance or prejudices like in the case of Feanor and his relatives ;) or other proud elf lords of the Elder Days)
dwarves love treasures and things made by hands more than nature and plants, elves on the contrary they love nature more (even though elves too like treasures and crafting, Sindar learned from them much in smithing weapons, Teleri liked silver and white jewels and Noldor were delighted in crafts making many beautiful things out of gold and jewels, yet they were less greedy, this interest in smithing, mining and many other crafts is what brings Noldor and dwarves together, elves just love to make things, just like dwarves so it's a natural common ground, Elrond is said to not approve the desire for gold of the dwarves but still he was eager to help Thorin's company)
the elves as immortal and wise may seem arrogant to mortals, in fact elves many times make fun of "clumsy mortals", some of the elves are almost childlike, joking laughing, mocking and sarcstic, while dwarves are more serious, obsessivelly interested in their self esteem, proud and stubborn (pride and stubborness are not exclusively dwarvish of course, many elves are like that, Sam described the elves as "Some like kings, terrible and splendid; and some as merry as children." Dwarves also can appreciate lighter side of life, enjoying feasting, songs and music, untarnished beauty of earth itself, caves, stones, mountains. Noldor too appreciated stones and beautiful caves, like the caves near the river Narog which became Nargothrond. You see though seemingly antagonistic, each race has similar characteristics, in similar proportions :)
elves think often that dwarves are "grasping and ungracious", dwarves think that the elves are strange and aloof (also dwarves did not give their trust easily as a secretive folk)
power and corruption of the Seven Rings (quite possible, even though Sauron could not directly control them because of their strong resistant nature, he could influence them to greed, anger and after a while to madness which could affect the relations between the peoples of elves and dwarves, if their kings were so unreliable and emotionally unstable, and of course Sauron's policy of dividing his enemies too took its effect, even elves were not wholly trusted by men in late Third Age if Lothlorien was rumored to be evil place and Galadriel was considered a sorceress)
As for historic events, ohh boy lots of them (some already mentioned):
Sindar hunting the Petty Dwarves thinking them to be strange beasts (Petty Dwarves were exiles and outcasts from other dwarven communities, lesser in size, aggressive, they attacked elves first taking them for intruders in their lands, Petty Dwarves came to Beleriand first according to their tales and did not tolerate the newcoming elf hosts)
murder of king Thingol and sacking of Doriath
dwarves not helping the Noldor in Eregion during War of Elves and Sauron (it's my own speculation, because we don't have any accounts on their involvement beside the line in Tale of Years, "1697 Eregion laid waste. Death of Celebrimbor. The gates of Moria are shut. Elrond retreats with remnant of the Noldor and founds the refuge of Imladris."). Apparently some military help was lent to elves of Eregion after all, but it wasn't enough to defend the land against Sauron's forces
dwarves releasing Balrog (many elves of Lothlorien escaped south to Edhellond, this means that Balrog must have made an impact on the inhabitants of the region, elves thought it was dwarves fault and their greed for mithril, Tolkien explained that Balrog was already awakened by Sauron's rising in power and dwarves only reached his hiding place unleashing him)
dwarves cutting down woods in the valley of Nanduhirion (dwarvish Azanulbizar, after the battle with orcs they used the wood to burn bodies of their countless fallen, Wood Elves probably took this not overly enthusiastic, even though in The Hobbit we have accounts of elves cutting trees in Mirkwood in need and many other elvish communities needed wood for various tasks, from lighting fires to building homes)
events of The Hobbit, imprisonment of Thorin and company in halls of Thranduil (here in the book we have also mysterious reference to "wars for treasures" which Elvenking Thranduil waged with dwarves, question of price for some unwrought gold and jewels was mentioned, there are speculations that it's only a hidden hint to Thingol and First Age but who knows, we don't know much about history of Woodland Realm in Mirkwood, maybe they did have wars with the dwarves in their time, Tolkien make lots of suggestions to such mysterious events that not appear in any official timeline, adding to the depth of his imaginary world)
So the reasons are two-sided, two races are guilty of their hard relations, but thanks to the course of story in Lotr books we see that they can overcame their strife and see past the things that divide them and become good friends.
In the case of destruction of dwarf-kingdom of Erebor by Smaug, it's possible that elves sent no help because they were not even asked for it. Many dwarves were slaughtered by dragon and survivors immediately fled to Iron Hills to live with Dain or wandered alone in Rhovanion until they gathered after some time, following Thrór to Dunland and later Thrain to Blue Mountains. Mirkwood elves were more isolated and constantly defended themselves from darkness spreading all over the forest and especially from evil creatures that inhabited it. Secondly, no army even of beings as fast and agile as elves could not arrive at Erebor so quick after Smaug's attack, it was several days journey from Long Lake and Mirkwood lays some miles further from it. After Smaug's death dwarves had lots of time to prepare themselves for the coming armies (not to mention that Thranduil was asked to help Lake-men of destroyed Esgaroth and this delayed them even more).
And in the War of Dwarves and Orcs they did not asked for help treating it as a matter of honor (so movie Thorin accusations towards elves, that they betrayed his grandfather or that they looked on and did nothing when orcs plundered Moria made me laugh, but hey it's a Peter Jackson vision not Tolkien's).
Tolkien says that the main catalyst was the Nauglamir and its resulting rui of Doriath and murder of Thingol in the first age.
The exact circumstances of this event are covered in other answers, but here is a quote of Tolkien explicitly saying that it's what led to the animosity thousands of years later.
But in far distant days the Dwarves were secretive and had few dealings with the Elves. In the West at the end of the First Age the dealings of the Dwarves of the Ered Lindon with King Thingol ended in disaster and the ruin of Doriath, the memory of which still poisoned the relations of Elves and Dwarves in after ages.
The Peoples of Middle-earth - "Of Dwarves and Men"
Read Tolkien's "Silmarrillion." When Elves marched against Morgoth Dwarves (who had allied with the Elves) brought up the rear -and when the dark forces engaged the Elves the Dwarves attacked theElves' flank, betraying their alliance. If memory serves that is the genesis of LOTR lore. Dwarves started it!