SHORT ANSWER: Yes, there will be animals in heaven because they are part of God's creation. However, they will not include the pets we have come to love here on earth.
LONG ANSWER: To start off, the Bible doesn't say anything explicitly about whether animals go to heaven. However, there is ample support in the Bible to conclude that YES there will be animals in heaven.
The traditional view is that there will not be animals in heaven because they do not have souls like humans do. However, this is only partially correct. The Bible is not clear about whether animals have a soul, primarily because there is a lot of debate about what the soul actually is. So, instead, I'll focus on what the Bible is clear about.
Humans and animals are SIMILAR in that we both are part of God's good creation and have been given the "breath of life." In creating man,
"Then the LORD God formed a man from the dust of the ground and breathed into his nostrils the breath of life, and the man became a living being." (Gen. 2:7)
In referring to the animals,
"And to all the beasts of the earth and all the birds in the sky and all the creatures that move along the ground—everything that has the breath of life in it—I give every green plant for food." (Gen. 1:30)
So, in this way, humans and animals are the same, in that we share the divine life force that God alone gives to all creatures at birth, sustains throughout their life, and takes back at death. Some argue that this is what we should call the soul.
Humans and animals are DIFFERENT in that humans are created in the image and likeness of God, while animals are not. In Gen 1:26-27, it says
"Then God said, 'Let us make mankind in our image, in our likeness…So God created mankind in his own image, in the image of God he created them; male and female he created them."
Because of the repetition of the phrase in these two verses, it seems to be a pretty big deal. Because we are like God, we are spiritual beings, in additional to being physical beings. Some argue that this is the basis of humans having a soul.
So, where do we go from here? Would you believe that the whole question of animals having souls doesn't really matter? It's because all of God's creation (including animals) will be redeemed in the end. Just as God's natural creation was also cursed by the sin of the Fall (see Gen 3:14-24), it will also be redeemed and restored. My favorite passage that talks about this is Rom. 8:16-35:
"I consider that our present sufferings are not worth comparing with the glory that will be revealed in us. For the creation waits in eager expectation for the children of God to be revealed. For the creation was subjected to frustration, not by its own choice, but by the will of the one who subjected it, in hope that the creation itself will be liberated from its bondage to decay and brought into the freedom and glory of the children of God. We know that the whole creation has been groaning as in the pains of childbirth right up to the present time. Not only so, but we ourselves, who have the firstfruits of the Spirit, groan inwardly as we wait eagerly for our adoption to sonship, the redemption of our bodies. For in this hope we were saved. But hope that is seen is no hope at all. Who hopes for what they already have? But if we hope for what we do not yet have, we wait for it patiently."
Because animals were part of God's original creation, I believe they will be part of the new, restored, redeemed creation (the new earth mentioned in Rev. 21:1).
HOWEVER, a question that usually goes along with this is "Will my pet Fluffy be in heaven with me?" Having a Yorkies that I love dearly, I've even wondered about this one. The simple answer to this is that we don't know. While there will be animals in heaven, we just don't know if they are the same ones that we've had as pets.
But we shouldn't worry too much about it question. Why? 'Cause Jesus told us not to. Would you believe that some people actually approached Jesus once and asked him a similar question? But this one wasn't about pets, it was about wives. In Matt. 22:23-30, some religious leaders asked Jesus about marriage after the resurrection. Jesus replied by basically saying that there won't be marriage in heaven. So, if we aren't going to claiming husbands and wives in heaven, I doubt we're going to be claiming pets either. And yet, it won't bother or sadden us because the glory and joy of heaven will be so complete that the thought of missing a pet (or a wife) won't even exist.
Pretty cool, huh?