It is wrong to kill human beings. Human beings are animals. So it is wrong to kill animals.

“Human beings” is a smaller concept than "animals". 

It is wrong to read bad books.

Bad books are books.

Therefore, it is wrong to read books. 

P1) It is wrong to kill humans.

P2) Humans are animals.

P3) It is wrong to kill animals.

The problem lies in P3. If humans are animals, it does not follow that all animals are human. Thus, given the constraints of P1, it does not follow that it is wrong to kill animals in general, only some of them (namely, the humans). Yet, if someone read "it is wrong to kill animals", they'd implicate that this meant all animals (if I said, "it is wrong to eat meat", you'd implicate that this refers to all meat).

This seems like fallacy of composition though the specific fallacy name, admittedly, is not clear in my head at the moment. My reasoning is that humans (A) are a subset of animals (B), and because A has X moral property ("should not be killed"), the speaker infers that B also has X moral property.