In order to answer the question we must first define heaven. According to the Catechism, heaven is a “state of supreme, definitive happiness” that involves a perfect “communion of life and love with the Trinity, with the Virgin Mary, the angels and all the blessed” (CCC 1024).
This definitive state of human happiness and communion of life and love presupposes the powers of rational knowledge and love, which animals do not have. Therefore, no non-human animal would be able to experience “heaven” as defined.
Perhaps our pets won’t exist in heaven in the sense of experiencing the Beatific Vision, but could they simply exist in the “new heaven and new earth” (see CCC 1043; 2 Pet. 3:13; Rev. 21:1)?
We know from the natural light of human reason that the pets we have now will not exist in the new heaven and new earth. The souls that animals have are entirely dependent on their material makeup. As such, they do not continue to exist after bodily death as human souls do. Therefore, all animals that die before the advent of Christ’s second coming will not exist in the new heaven and new earth.
And there couldn’t be any sort of resurrection of these animals since resurrection presupposes a continued existence of some sort. Fido can’t be resurrected and brought into existence from nothing at the same time.
But to the question of whether animals in general will exist in the new heaven and new earth, the Church has never given any definitive teaching on this question. Any answer provided would be in the realm of speculative theology.
There are arguments on both sides. Those in favor argue that there is nothing intrinsic to the nature of God or animals that would prohibit God from creating new creatures and miraculously keeping them in existence. Just as God will preserve other material things from corruption, including the human body, he could very well create new non-human animals and preserve them from corruption.
Those against the idea of non-human animals existing in the new creation argue that there will be no need for them. The perfection in joy that humans will have in God and one another seems to preclude the need for the affection pets give us and the need to use animals in general for achieving certain natural goods.
The bottom line is that we know for sure that whatever pets die before the advent of the new creation, they will not be there. But concerning the question of animals in general, that is ultimately unanswerable given the contents of divine revelation.
Here is a great article on this subject written by my good friend and colleague Todd Aglialoro “The Fidoist Heresy.”