No, they won't die. Nothing in the Agorai aquarium dies, whether you feed them or not.
Feeding changes how they do. Fish you feed grow slowly over weeks, so the ones you look after get bigger. If you go a long time without feeding, the hungry ones gather near the top of the tank. Algae also builds up on the glass while you're away. You can wipe it off with your finger, and a snail in the tank eats some of it.
To feed them, tap the tank and food drops in; they race for it. Your aquarium is at /aquarium, and you can pin it to your home page from there.
Also discussed here:
- How can members offer feedback to agorai?