In Lord of the Rings, the rings preserve what you desire the most. The Elves have three rings preserving their kind's well-being, as well as preserving Rivendell. If the One Ring is destroyed, and this takes the Elves' three rings down with it, what motive do the Elves have to help?
This in answered in the Lord of the Rings chapter The Council of Elrond, during a discussion between Glóin, Elrond and Glorfindel (with my emphasis):
'But what then would happen, if the Ruling Ring were destroyed as you counsel?' asked Glóin.
'We know not for certain,' answered Elrond sadly. 'Some hope that the Three Rings, which Sauron has never touched, would then become free, and their rulers might heal the hurts of the world that he has wrought. But maybe when the One has gone, the Three will fail, and many fair things will fade and be forgotten. That is my belief.'
'Yet all the Elves are willing to endure this chance,' said Glorfindel 'if by it the power of Sauron may be broken, and the fear of his dominion be taken away for ever.'
This is confirmed by Galadriel's words to Frodo later on in the chapter The Mirror of Galadriel (again with my emphasis):
The love of the Elves for their land and their works is deeper than the deeps of the Sea, and their regret is undying and cannot ever wholly be assuaged. Yet they will cast all away rather than submit to Sauron: for they know him now.
The Elves' motive is therefore to achieve the destruction of Sauron's power, and loss of the power of their own Rings is a tradeoff that's acceptable to them in exchange for that.
Relevant to this are Elrond's words elsewhere in The Council of Elrond, where they discuss the possibility of using the Ring against Sauron:
If any of the Wise should with this Ring overthrow the Lord of Mordor, using his own arts, he would then set himself on Sauron's throne, and yet another Dark Lord would appear. And that is another reason why the Ring should be destroyed: as long as it is in the world it will be a danger even to the Wise. For nothing is evil in the beginning. Even Sauron was not so. I fear to take the Ring to hide it. I will not take the Ring to wield it.
This highlights that in the end the only realistic option was to destroy the Ring.
The three elven rings were much different then the other rings of power. The 7 rings of dwarves and 9 rings of men had one major similarity, their main purpose was to enhance the power of the bearer. Although forged by Cerebrimbor, the three elven rings served the main purpose to heal and preserve.
The three elven rings -
Narya - Passed from Gil-Galad to Cirdan to Gandalf, helped Gandalf to inspire others to fight against evil and imbue hope into others' minds.
Nenya - Worn by Galadriel and used to protect Lothlorien.
Vilya - Passed from Gil-Galad to Elrond, the mightiest of healers of his time. It was responsible for helping Elrond protect Rivendell and also possibly for his magical abilities (although he may have some by birth).
The elven lords, when they were first given the rings by Celebrimbor, wore them but when Sauron tried to corrupt their minds they realized it and hid them. Only after when the one-ring was cut off from Sauron's finger, they started using it to rebuild safe-haven's for all those elves still remaining behind (when the war was won by Last alliance of Men and Elves most elves decided to leave for Valinor). When Sauron's power began to rise again they felt that it will not only destroy men and dwarves but also bring great harm to elven populace. So, for self preservation and also to honor the great alliance with men they joined the fight against Sauron (i.e. destruction of one-ring).
It is rightfully said, that once the One-Ring was destroyed the power of the three elven rings also withered away. But then again, by that time most of the elves had decided to return to undying lands any way. So the purpose of the three rings (to create pockets of elven safe haven such as Lothlorien or Rivendell and also to inspire the oppressed) were properly served.
Their only options were-
Help destroy the Ring.
Abandon Middle-Earth and go to Valinor.
Stay and do nothing as they watch Sauron conquer all of Middle-Earth.
There was no other way to defeat Sauron and the Rings were not going to protect the Elven realms when Sauron put his full power against them. The time of the Elves was on the verge of ending one way or another.
The Three Rings were going to become useless no matter what else happened. If the One Ring was destroyed, the Three would lose their powers and become ordinary pieces of jewelry. If Sauron found the One Ring, the Elves would be forced to stop using the Three, because using them would lead to Sauron controlling the Elves who wore them.
This means that the Elves had to choose: Help the Fellowship destroy the One Ring, thereby saving the world but losing the powers bestowed upon them by the Three; or do nothing, see the world being destroyed, and still lose their powers because they could no longer use the Three safely; or see the world being destroyed, keep using the Three, and become slaves to Sauron.
Given these options, the Elves clearly made the right decision.
If Sauron regained the Ring the elves would've been utterly ruined as Gandalf said "all that has been wrought by those who wield the Three will turn to their undoing and their minds and hearts will become revealed to Sauron if he regains the One. It would be be better if the Three had never been". I may be wrong but I've always thought that if Sauron regained his Ring then the 3 elvish ringbearers would become like the Nazgul, completely enslaved to Sauron.
If Sauron gets his ring back. Elves would deal with him again. They might be thinking that to hit him while he is at his weakest and with the hardest hit (which is to destroy the ring) is wise. They might also be trying to finish what they started. Since humans took the ring and made the bearer corrupt, they might not want another incident like that. Now the ring has been found, they have the chance to finish the job.