I suppose most Thomists would say that animals do not go to heaven, given that they do not possess immortal souls. This somewhat harsh response is often softened with the assertion that they are not entirely gone in that other animals (like dogs) share their substantial form. Others would say that an animal, like your favorite dog, continues to exist as an idea in the mind of God.
C.S. Lewis remarked that canine loyalty and affection oftentimes put human fidelity and friendship to shame. Because of this he thought that maybe dogs would be allowed to join their masters in heaven. Critics contend that this is just another instance of over-blown English sentimentality.
In any case, there is a growing concensus that the outer circle of hell is patroled by cats.