Will it hurt cats to drink apple juice?

It's not toxic to them, so if your cat gets into your Apple juice it doesn't necessarily warrant an immediate trip to the vet. (Grape Juice, though, DOES seem to be poisonous to both cats and dogs, so it's not safe to assume that just because you can drink it means that your cat can.)

It's not a good idea to let them drink too much of it, because Apple juice has a lot more sugar in it than is really good for cats; cats lack the tastebuds for sweet things anyway, so the sweetness is not why they're drinking it and they'd probably be just as happy with some water with a few drops of cider vinegar in it. Also, Apple juice causes ... let's call it "intestinal distress" ... in some people, so you might want to monitor the litter box and stop allowing your cat to drink Apple juice if it seems to be causing any problems.