Based on evidence of food we find throughout the game (thank you, Mr.Verto), we can make a perfectly logical assumption that Hylians more or less have to eat in order to live, just like the rest of us (but, as Zleda mentioned, they reason they never eat when anyone is watching is probably because they are part of the gaming experience of all of Nintendo's valued customers).
We don't regularly see Hylians eat, so we can't completely rule out the possibility that, as ENA-three! suggested, nobody eats. But that would not explain all of the evidence of food.
Perhaps there is a compromise in here somewhere--perhaps Hylians CAN eat but don't necessarily have to. But that would be kind of odd, and I don't think Nintendo would purposefully create a world full of people like that and then not say anything about it.
So if Hylians have to eat, I still consider it a mystery how they do it when the world is covered in Twilight. Even if a Hylian food has a Twilight Realm equivalent, as Wolf Sage and a few others were saying, I can't see how any Hylian could eat something that comes from a world that they can't even exist in.
LinkLover suggested that perhaps ghosts don't have to eat, but the game said that nobody could tell that they'd been turned into a ghost.
Now, what about Midna and the Twili?
We never see tables or food in the Twilight Realm. But Twili move around and even do magic, so they have to get the energy it takes to do that from somewhere. Since Twili have teeth, it isn't unreasonable to assume that they can eat (maybe they like bread XD). We see birds in the Twilight Realm, so a variety of animal life probably exists there.
Is eating the ONLY way a Twili can get energy? It's impossible to say since they can do magic and magic is kind of a wild card. Mybe Midna "eats" by... I don't know, pulling energy out of the universe through magic when she's in the World of Light. But that arrangement seems about as awkward as the idea that Hylians don't need food to survive.
Ok, carry on.