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:
They collaborated in works like the creation of the fortress of Menegroth in Doriath:
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:
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:
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.
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 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:
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:
So basically, the elves and dwarves were rival creations, destined to be at odds from the very beginning.
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!
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 it's 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 :).
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:
Now, the other Dwarves did despise the Petty Dwarves - but:
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. ;)
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!