Whenever I hop onto my horse, it will sometimes take damage, and will sometimes even die, despite being at full health. Sometimes my player will take a little hit of damage as well. Also, horses will sometimes take a bit of damage when I first equip a saddle and armor. It might be from the armor, or the saddle, or either, I'm not sure. It can happen on any kind of ground, including half slabs and stairs. It can happen when the horse is leashed to a fence, or when it is free roaming. I have tried leashing a horse and pulling it a few blocks from where it was when the chunk loaded, and it still happened.

I haven't tested it in water. I haven't had it happen with a Donkey, Mule, Llama, or Pig, but I don't ride those often.