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.
When discussion the nature of death of plants and animals, Tolkien says that elves will kill animals and plants for substance, and that while they may not like to do so, it is "their nature to be fed" and that they "have a right to live". So they need to eat to surivive.
Some of the Eldar[=elves] (and some Men) eschew the slaying of kelvar[=animals] to use their bodies as meat, feeling that these bodies, resembling in different degrees their own, are in some way too near akin. (Yet none of the Eldar hold that the eating of flesh, not being the flesh of the Incarnate and hallowed by the indwelling of the fëa[=soul], is sinful or against the will of Eru.) But even so they must kill and eat olvar[=plants] or die; for it is their nature to be fed, as to their hröar[=body], by living things corporeal, and things have a right to live according to their nature. Yet violence is done to the olvar (which have a kinship with the bodies of the Incarnate, be it remote), and these are denied the fulfilment of their own lives and final shapes. Therefore we must hold that the Incarnate belong by nature to Arda Marred and to a world in which death, and death by the violence of others, is accepted. Neither Elves nor Men eat willingly things that have not died by violence.
The Nature of Middle-earth - "Death"
Frodo says this after being rescued in Cirith Ungol
“No, they [orcs] eat and drink, Sam The Shadow that bred them can only mock, it cannot make: not real things of its own. I don't think it gave life to the orcs, it only ruined them and twisted them; and if they are to live at all, they have to live like other living creatures."
While it's not completely definitive, this strongly suggests Frodo believes Elves need to eat. He's spent months in Elven settlements and even more time with Legolas in particular, so there's a good chance he'd know.