Yesterday afternoon I buried a cat. A friend of ours, a single mom, had to put the family cat down, and the kids wanted it to be buried in the back yard. So I dug a hole (no easy feat in Virginia red clay) and we laid the cat to rest.
Although the kids did not ask me "is our cat in heaven now?", I believe that we very well may see beloved pets in the Kingdom of God. I can't speak formally for the Church's position on this, to the best of my research she doesn't have one. It has never come up as a question that needed a definitive answer. All the same, taking what I do know about the life of the world to come, I'd like to lay out my case for why we may have pets in the kingdom.
The Church confesses in the Nicene Creed that God created all things visible and invisible, which includes pets. Genesis (1:25) tells us that when God created the beasts of the field, He saw that it was good. The Church teaches that Jesus became Incarnated to redeem not only fallen humanity, but all of creation as well.
Many of the prayers of the Church (St. John Chrysostom's liturgy is full of these) which ask for God's mercy include the statement "For You are good and love mankind" It is in God's nature to love us, and do good for us.
So, God made animals, and animals are good, and the animal kingdom will be redeemed along with mankind at the Second Coming when Christ makes all things new. The dead in Christ will be raised and their bodies restored and glorified. It is not at all a stretch to believe that beloved family pets may be so raised as well. This would be a cause for much joy among His people, and our loving Father loves to give good gifts to His children.
This is my reasoning which leads me to believe that we may (I cannot say will, for I am no prophet, nor do I read God's mind) have our pets with us in the Kingdom of God.