Yes, they'll die. You can resurrect them through a quest, but that can be expensive.
If you put your pet in your rancher (even if you don't price it) or your showroom, though, they won't starve. They won't age either, but if for some reason you have to go somewhere, you can store them and ensure that they live.
It might be worth mentioning: you can also load your food pen with food, so your profile creatu will be fed automatically in your absence. Just make sure you put enough food in there for the time you plan to be gone. It's also convenient to do if you still want to play but don't want to be bothered with feeding them everyday.
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