Discussion on this recently and on a search it appears the RCC does not have a definitive stance on this, although some claim they do…even recent popes such as John Paul and Francis have spoken out supposedly in favor of this being true. Thoughts?
My thought is that the Bible speaks of a new Heaven and a new Earth. If the new Earth has no animals, I would find it a strange place. I like animals, generally. My wife grew up on a farm. I hope there are animals on the new Earth.
I hope there are animals too, it would be nice to find my dogs there.
I have a co-worker who LOVES animals (probably more than she loves people) and she was devastated to find out that some Christians do not believe that there will be no animals in Heaven. I joked that I hope there is, because how else would we be able to eat fried chicken ; )
There was an early episode of “The Simpsons” where Bart and Lisa go to Sunday School and Lisa is told that animals do not go to Heaven and she left very disheartened.
From a theological perspective, animals are said to have limited souls and for this reason, may or may not be in Heaven.
I’ve heard others say that if God knows that animals will make you happy in the life thereafter, then He will give you animals in Heaven.
I look at it slightly differently. What if one’s love for animals is greater than their love for God? And what if one would not desire to be with God eternally in Heaven if there are no animals there?
What if, on the other hand, one’s hatred of animals would make Heaven with animals Hell for them?
And are we talking all animals, only cute, domesticated animals, or animal souls whom we ask God to save?
Animals are without sin, so no Hell for them. 
As for which animals would go there, yeah that is a slippery slope…like if so, where is the line drawn? Do reptiles get in? Fish? Insects (ugh I hope not)?