Our pets are precious to us. And sometimes it is difficult to imagine heaven without them. One Canadian broadcaster even wrote, "I was quite shaken by this revelation: an afterworld that deprived me of my dog seemed to me less than heaven."
Though we can appreciate that man's attachment to his pet, we have to look to the Bible for an answer to the question of animals in heaven. Obviously, the Bible doesn't give a direct answer. But it does provide information about heaven and animals to guide us to a better-informed discussion of the matter.
The Bible teaches that people are unique beings on this earth. We were created in the image of God (Gen. 1:27) and given dominion over the earth-including the animal world. Psalm 8:4-8 says,
What is man that You take thought of him,
And the son of man that You care for him?
Yet You have made him a little lower than God,
And You have crowned him with glory and majesty!
You make him to rule over the works of Your hands;
You have put all things under his feet,
All sheep and oxen,
And also the beasts of the field,
The birds of the heavens and the fish of the sea,
Whatever passes through the paths of the seas.
Since we are uniquely provided with an exclusive role on earth, we can expect differences in how God shapes our future. Redeemed humanity will be in heaven praising God, particularly for His redemption provided in Christ. Unredeemed humanity will exist in eternal torment for their sinful disobedience and rejection of Christ. We have moral souls accountable to God even after our bodies die (Heb. 9:27).
Animals don't have a moral dimension to their existence. They don't sin against God and cannot understand the good news about Jesus, so they cannot become Christians. Animals are not spiritual beings like humans are. So animals do not go to heaven in the way that humans do by believing in Jesus and gaining eternal life.
That doesn't necessarily mean, however, that there will be no animals in the future. Isaiah 11:6-9 says that they will be on the earth when Christ comes back to rule for a thousand years. And when God creates the new heaven and earth (see Rev. 21 and 22), He may choose to create some animals for His glory and our enjoyment. But since the Bible doesn't say either way, we just have to trust God to provide everything we need for happiness in heaven.
For more information on heaven, see John's book, The Glory of Heaven.