On the one hand, they are physically very similar to humans, and the only crucial difference is that they don't age or fall ill. This, as well as frequent mention of food being eaten by elves, gives the idea that they do need to eat. On the other hand, Tolkien mentions elves that had been lost in the ice fields on their journey from Valinor to the Middle-Earth, and he speculates that they still wander there being lost. As there is practically no food to be found there, this seems to suggest that elves don't need food to survive.
It seems so, but there is no clear evidence either way. The Silmarillion does hint that Elves can starve and die of extreme cold, and that perhaps some did while crossing the Helcaraxe. The "Ask Middle-earth" blog covered this.
Tolkien states right out that “the Elves die not till the world dies, unless they are slain or waste in grief (and to both these seeming deaths they are subject).” So possible elf deaths fall into two categories: those that are slain, and those that die of grief...
Though nothing is said about it directly, it’s also suggested that elves could die of starvation or extreme cold. There’s a few references made to this in The Silmarillion, though never of an elf actually dying of these causes. Either way, it’d take some serious starvation or extreme cold to do it, as elves are much much hardier than men.
As for the “waste in grief” option, that’s a bit more complicated, just because it’s vague (typical for Tolkien.) Luthien, when Beren died (the first time), was said to die of grief and broken heart. But then there’s the interesting case of rape. It’s said in “Laws and Customs of the Eldar” that, in the case of rape “one so forced would have rejected bodily life and passed to Mandos.” The only way to avoid this was to travel to Valinor immediately, like Celebrian did. So it seems to me that when something horrible (like a broken heart or rape, based on the examples Tolkien gives us) happens elves can literally lose the will to live, and so they die.
It seems likely that an Elf wouldn't starve very quickly, though - the Elf Voronwë spent seven years lost at sea and didn't starve; although we might assume that he had plenty of fish to eat, the story (in Unfinished Tales) never says so.
Elves are extremely concerned about preserving and healing the world, so when we read about them hunting (which we do, fairly frequently), it is almost impossible to imagine them doing so unless they need to eat to survive.
It appears that at least two Elves are known to have starved to death: the twin brothers Elurín and Eluréd. They were the sons of Dior, who was in turn the son of Beren and Lúthien. The sons of Fëanor attacked Dior's kingdom, killed Dior, his wife, and many others, and sent the boys into the woods to starve. Maedhros, one of the less horrible sons of Fëanor, later repented this act, and spent quite a bit of time trying to find and rescue the boys, but to no avail. Christopher Tolkien has confirmed that they died shortly after being sent out, when they were only 7 years old.