The difference between consent and lack of consent is pretty fundamental. It would involve not only a change in marriage law, but a fundamental shift in legal understanding, to where it no longer mattered whether someone gave consent or not. In that system, rape would be legal, as would theft, murder, and probably everything else. Consent is the fundamental concept here. Animals and children, by our definition, can't consent (and it's very obvious why - there's really no need to make this more complicated than it is).
Do we require an animal's consent before it's sent to the slaughterhouse? Must a dog give it's consent before being spayed or neutered?
Marriage between man, woman and animal would require absolutely no shift in legal understanding. It would require only another shift in the definition of marriage.
I'd assume most people would want to ensure that the animal isn't being abused and existing animal cruelty laws could be left unchanged. But if LeRoy (with his jar of peanut butter) and his dog want to get married, who are you to say no?