I'd think it's best to put in the fridge... that said, my cats get fed at around 10 pm and we don't change their food until the next day, whenever they start asking for food... which isn't always in the morning. If there's still some left at that point we throw it out, however, they're free to eat it until the time they get fed the next day and like I said, it varies.
Meanwhile if my dog doesn't eat her wet food she just gets it again the next day with some extra added to it (we only do this one day though, then it gets thrown out). Sometimes it gets put in the fridge, especially in the summer, other times we leave it out the whole time - 24 hours.
We've never had any problems doing this with them, animal disgestive systems are built differently than people.