Although there are references to some specific animals in the Kingdom of God, there is nowhere in scripture that says that animals go to heaven when they die. This is because the Bible is written for human redemption, and not animals, and therefore is silent on this particular issue. There are segments of scripture that say that all of creation shall be redeemed. But these refer to the renewal that will take place when God creates a new heaven and a new earth. For an animal to die and then go to heaven it must have a soul. Souls are eternal, and they continue to exist after the passing of the physical body. And if an animal has a soul, then there must be a means of salvation for it to enter heaven, for every soul on earth is subject to sin. And if there is salvation, then pets must therefore have a choice in the matter; and no one wants to think about cute little fluffy being burned to nubbin in hell because she made the wrong choice. (I had a cat once that I'm pretty sure was possessed!)
1Cor 15:35,39 "But someone will ask, "How are the dead raised? With what kind of body do they come? For not all flesh is the same, but there is one kind for humans, another for animals..." Paul continues to say that man will be raised with a new body, one that is imperishable. The Bible doesn't speak of animals having a soul. And it is a man's soul that goes on into eternity when his body has died.
I know many people get attached to animals like they would a family member. And it is really hard to think about being separated from the pet. So it's a natural response to wonder if they will go to heaven when they die. What we must realize is that death is a part of this life, and separation with it. God gives us animals and pets in this life so we can enjoy this life here on earth. They are gifts for the "now" just as we will be given new gifts "then" in our ressurrected state.
The Bible does speak of animals being in heaven, however. And these are animals not ressurected from earth, but created by God for that realm. In Revelation 19, it tells how the armies of heaven ride out on horses following Jesus.
In the new Millenium, when we rule with Christ on earth, it speaks of
Isa 65:25 "The wolf and the lamb shall graze together; the lion shall eat straw like the ox, and dust shall be the serpent's food. They shall not hurt or destroy in all my holy mountain," says the LORD.
Ultimately, the Bible is silent on the issue. But if you are one of those tied to pets, I'm sure God has something even better in store for us when we enter our heavenly home than we could ever imagine.