This answer to applying sunscreen to dogs says that you shouldn't apply human sunscreen because it contains zinc oxide and that's bad for dogs. What about it is bad, and what would happen if a dog did come in contact with it?
The primary reason for concern would be zinc toxicosis. This is basically the formation of zinc salts in the stomach acid which are toxic and can lead to diarrhea, lethargy, seizures, and more. It's quite dangerous and can require surgical intervention to treat.
Common sources of zinc poisoning in dogs and cats include the Lincoln penny(!) and zinc-oxide creams. Dogs are more likely to get this problem because they more freely eat whats around them, but using human sunscreen on a dog or cat is very likely to result in them attempting to lick it off and, well, poison themselves.