"As far as I can tell, nothing. Being left alone for an entire arena doesn't affect them. All I've ever had to do for mine was give it affection and I'm not sure that's something that it needs so much as something it enjoys."

So they're perfect pets for tributes, really. They won't starve and die if they're left in one's room for the length of an arena (or the length of a stay in a peacekeeper cell). If they need water they seem to be able to find it fairly well on their own.