Do I need to eat?

In Die2nite I've noticed several other people dying from dehydration, but haven't seen anyone die of hunger yet. I myself have been 3 days without food.

Will I eventually die of hunger, and will there be any warning?

No, hunger is not implemented in Die2Nite. For thirst, you get a notification the night before you die of dehydration that you need to drink, giving you ample opportunity to do so.

Eating is only for restoring AP, not for staying alive.

Realistically, humans can survive without food for a few weeks if necessary (won't be pleasant though). So, as alexanderpas mentioned, you will probably die of zombies much earlier.