The worst that can happen is that your pet gets impregnated, births an alien child in town, who then proceeds to impregnate the rest of the town, turning it into a wasteland filled with aliens who will attack you on sight.
Now, that's not a very likely scenario, but it could certainly happen. When you don't feed your pet, they will drink from fountains, and that has a host of things that can happen, one of which is the above. Other things are your pet randomly falling in and drowning (and thereby causing you to pay for their resurrection). Whatever can happen from drinking from a fountain, they might get it.
The other bad thing is that they will eat rotten food when hungry, too. Strangely, they won't eat any rotten food they're carrying, but if you drop some on the ground, they go for it like starving wolves.
Now, it's not ALL bad. They won't starve at all, so you don't need to worry about your pet keeling over from hunger. But there are good things that can happen from eating, and those apply to your pet as well. If you want them to train certain stats, make sure to give them food that does so. And keeping them full ensures that they won't eat that corpse that just dropped, that gives you the training YOU want.