1 - Only someone who hasn't had a dog would say this isn't a serious question. I say this not to demonstrate that dog-lovers are hopelessly sentimental and deluded, but just that there is an analogy between the way we feel about pets and the way we feel about other people. Of course, human relationships are deeper, more complex and, at least potentially, more satisfying (as well as being potentially much more painful). But our feelings for a pet have some similarities. In our house, over the years, there have been and still are, lots of animals. One of the many reasons that we keep them is that we think that learning to care for an animal, with all that entails, will help prepare our children to be better carers for their fellow human beings. Not that this love for the dog, cat or whatever it is, in any sense a "pretend" love. It is completely genuine.
This is important because the ability to love is one of the things that makes us human (one of the good things, that is). For the Christian, it is also one of the things that reflects God's image in us. God is Love [1 John 4:8], he loves us [John 3:16], and in making us in his image he has given us the ability to love him back and to love our fellow creatures. Clayboy is right to say that we can't be too specific about what heaven will be like. But we can be confident that one of the things that will last for ever is love - this is why I am sure that part of the Christian hope is that we will be re-united not just with God, but with each other, even if we don't know exactly what it will be like. And that's why people ask the question about their dog-because they love him.
2 - One of the attractive but little-understood things about the Christian faith is that it is a hope not just for the souls of men and women, but a hope for all creation. The Bible in fact has quite a bit to say about this. God's intention is to redeem everything he has made through the power of Christ [eg. Romans 8:18-21, Colossians 1:19-20]. That being so, there seems no reason to assume that dogs, cats, hamsters, budgies, etc. will not be included.
Of course, we can't know what the new creation will really be like. But I find myself thinking of C. S. Lewis's The Last Battle, where heaven is pictured as a place where we encounter the things (and people) we have known in this life, but in a newer and better form that is somehow the same, but more so. Lewis's idea may owe more to Plato than to the Bible, but this is perhaps one point where we might allow Plato to be in agreement (or at least not in disagreement) with the scriptures. Paul tells us that in the resurrection, we will retain our bodies, but in a transformed state [1 Corinthians 15]. This seems as good a way as any of understanding how God will re-create the rest of the universe - the same, but different, and better. So we would expect to recognise what we see there.
One of the consequences of owning lots of animals is that over the years, we have sadly had to commit the mortal remains of many pets, and each one was grieved over to some extent, even the fish from the tropical tank. In order to comfort the grieving, I have developed a good line in (Bishops look away now) animal funerals. This is not as daft as it sounds, as I do have some sense of theological integrity and this experimental liturgy had to make sense to me as much as it did to a 5-year old. So what I usually pray is "God will remember (insert name or, for small creatures, this species) in his new creation". And I pray it with some confidence.
3 - Sometimes we have to think about what lies behind the question. Heaven is assumed by most people, both within and outside the faith, to be an essential part of Christian belief. But, because the Bible doesn't give us a cut-and-dried picture of what it will be like, there are lots of different ways that are used to picture it, some of which, frankly, are quite depressing. A friend of mine likes to remember her Youth Group leader who was forever telling the young people that heaven would be one long party - to which she was thinking "count me out then, can't stand parties". Another popular illustration that could do with a little more thinking through is the one where we're told that heaven will be just like church, except going on for ever. Hmm...
Even if we stick to the Bible and talk about being gathered in the presence of God, or being free from sin, we can be in danger of making it sound like an eternal committee meeting or AA gathering.
When people say "will my dog go to heaven?", or a number of other similar questions, quite often what they're actually saying is "Do I want to go there myself?". Those of us who do believe that it is worth looking forward to need to take questions like this seriously. Of course we can say "we won't really understand how good it is until we get there", but most people need something to keep them going between now and then (I certainly do). How to explain heaven is always going to be a challenge, but making a comparison between the "pleasures for evermore" and simple earthly pleasures like having a dog is not such a bad place to start.