The Bible does not specifically speak to this subject. However, it does not say that there won't be animals in heaven. In fact, there is probably good reason to believe that there will be based upon the fact that when God created a perfect world in Genesis, a part of His creation included animals. So to reason that there will be animals when God re-creates the world (2 Peter 3:3-13) is not such a stretch.
However, that being a possibility, the Bible teaches that only human beings were created in God's image and have a soul that lives on after physical death. This is why the Bible places infinite value on human life and why God's law in the Old Testament demanded the death penalty for the murder of another human life. Animals were not created in God's image and do not, as such, have an eternal soul. Because human beings are created in God's image, we possess a sanctity of life unlike any other living thing in all creation. This is one reason why human sacrifice was strictly forbidden in the Old Testament. However, under the Old Covenant, animal sacrifice was God's prescribed way--for animals have no living soul.
Animals are a wonderful gift from God. They bring much joy into our lives, and that is part of God's design. I am thankful for the "pets" that I have had in my life and for the wonderful memories of them. My life has been enriched because of those experiences. However, I do not anticipate seeing those same pets again in heaven as I do friends and family members who have passed from this life to the next. As a personal opinion, I do believe that there will be animals in heaven and that we will enjoy pets who never grow old, never get sick, never go blind, never make a mess where they're not supposed to (!), and who never die. Our enjoyment of animals in heaven will be full, complete, and infinitely satisfying . . . as a part of God's design.