In my own theological tradition, we believe that those children who die in infancy are numbered among the redeemed. That is to say, we hope and have a certain level of confidence that God will be particularly gracious toward those who have never had the opportunity to be exposed to the gospel, such as infants or children who are too disabled to hear and understand.
The New Testament does not teach us this explicitly. It does tell us a lot about the character of God—about his mercy and his grace—and gives us every reason to have that kind of confidence in his dealings with children. Some will make a distinction between infants in general and those who are children of believers, the reason being that when God made a covenant with Abraham, he made it not only with Abraham, but with Abraham's descendants. In fact, as soon as God entered into that relationship with Abraham, he brought Isaac into it—when Isaac was still an infant and didn't have an understanding of what was going on. This is the reason, incidentally, that a large number of Christian bodies practice the baptism of infants; they believe that children of believers are to be incorporated into full membership in the church. We see this relationship within the family in biblical history.
We also see David's situation in the Old Testament when his infant child dies. Yet David is given the confidence that he will see that child again in heaven. That story of David and his dying child gives a tremendous consolation to parents who have lost infants to death.
Now the point that we have to make is that infants who die are given a special dispensation of the grace of God; it is not by their innocence but by God's grace that they are received into heaven. There are great controversies that hover over the doctrine of original sin. Lutherans disagree with Roman Catholics, who disagree in turn with Presbyterians, etc., on the scope and extent of what we call original sin. Original sin does not refer to the first sin that was committed, but rather to the result of that—the entrance of sin into the world so that all of us as human beings are born in a fallen state. We come into this world with a sin nature, and so the baby that dies, dies as a sinful child. And when that child is received into heaven, he is received by grace.