Since the Bible has nothing to say about dogs and cats going to Heaven, a philosophical approach will be necessary to answer this question. While it is clear in Scripture that humans are a unity of material and immaterial parts (2 Cor. 5:8), no such thing is said about animals. We must work from what we know of humans and animals through experience. The major factor that separates humans and animals is their ability to reason. Both have the same senses and both have awareness of their surroundings, but only humans think rationally, using logic. Humans can analyze data; animals can only perceive. Animals can be conditioned; humans can be taught ideas.
Why is this important? It illustrates the difference between the merely physical phenomenon of perception and the immateriality of reasoning. No scientist has been able to discover any link between the physical matter of the brain and immaterial thought. In fact, it would be impossible to do so since science can only study physical phenomena. Immaterial substances are outside of the scope of science. Animals can be explained purely in material terms whereas humans cannot since they possess minds--not merely brains that receive.
If, then, humans are found to have immaterial minds, then a part of them exists apart from the body and may be capable of surviving death. On the other hand, if an animal dies, there is nothing left of it to survive. Once life has left the matter of the animal’s body, it ceases to be a living thing altogether and becomes merely a carcass. So, it would seem that, barring any other evidence suggesting that there is an immaterial aspect to animals, they do not go to Heaven (that is, the Heaven which exists now).
Having established all of this, there is good reason to believe that there will be animals in the new earth (Is. 65:17, 25), which will be the eternal home of those who are saved (Rev. 21:1, 22:5). This will be a very physical place lived in by physical people with resurrected bodies enjoying the material pleasures of the new earth (Rev. 20:4, 1 Cor. 15:20-22, Is. 65:21-22). If no form (mind or soul) remains which may be reunited to matter (a body), then animals which formerly died won’t be resurrected either. If my reasoning thus far is correct, then only new animals will be made by God. Ultimately, everything that you love about your dog or cat is found in an infinitely more perfect form in God Himself, and He will be all you need and all you ever desired. If you are without any loved one (including animals) in the new heavens and earth, you will not feel any lack because God Himself will fill every desire of your heart perfectly as the source of all good things you seek.