The play alludes to dogs, horses, donkeys, birds, and foxes, among others. I will provide a few examples to give you an idea, and then you can track down others for yourself.
Dog allusions are frequent in the play; for example, Andrew says that he is "dog at a catch," to which Feste responds that "some dogs will catch well." Andrew also says that if he thought Malvolio was a Puritan, he would "beat him like a dog." Fabian compares Feste's request that he "not desire to see this letter" to "giv[ing him] a dog and in recompense desir[ing his] dog again."
There are a few other allusions to animals. In 2.3, Andrew responds to Maria's horse allusion—"my purpose is indeed a horse of that colour"—by comparing Malvolio to a donkey and stating: "your horse would now make him an ass." Feste refers to himself as "no fox" in 1.5. When he and Malvolio are speaking in the prison, they discuss how Pythagoras believed that someone might be reincarnated as a bird.
These examples should give you a start, and you can discover some more for yourself. You might consider searching for animal words in an online text of the play.