I'd like to think not. On The Walking Dead, it appears they do but I recall another movie where our heros strapped food to a dog and sent it across a street full of undead to supply a friend who was cut off from the main group and the dog was not attacked. Dogs could probably be trained as early warning animals, smelling/hearing approaching Zs, as messengers, etc. I'd kinda like that. Horses/mules/cattle would be handy. And the Zs wouldn't have et up all the wild game.
Seems to me part of the horror of zombies us us being toppled from the top of the food chain. I never see zombie raccons or schnauzers so the zombie virus does not seem to "turn" common animals other than humans. Zombies have no nutritional requirements so killing us and spreading the virus would seem to be their only motivations. I don't see other animals even being on their radar. Thoughts?