What happens in heaven if you remarry?

Let me add a different view to this.

It is evident that procreation and sexuality were part of God's original intention for us (Gen 1:26-27: they were already created as man and woman), Gen 2:22-24 show clearly that sexual relations were part (or were intented to be part) of our life in paradise. Therefore the argument that the New Creation will offer us so many wonders surpassing our imagination that we would not want or need sexual relations anymore are hard to hold up against Gen 1 and 2. I would rather argue that proper sexual relations are part of these wonders and child birth with no pain and damage likewise.

Marriage is strongly linked with inheritance: The eldest son gets half of the father's possession. So it was crucial to ensure that the right man gets that blessing (cf. the confusion, violence, etc. caused from Jacob/Esau, Gen 27). A system in which you had a certain probability that a specific child was a specific man's child (remember: mater semper certa, pater semper incertus) made this issue a whole lot easier. That is also the reason for the draconian punishments on adultery: It disturbed family trees and inheritance lines for generations to come.

Infertility is a consequence of the Fall of Man. Therefore it cannot exist in the New Creation as our bodies will be restored to their original perfection (cf. Gen 1:28, Gen 30:1, Ex 23:26, Dtn 7:14, Psalm 113:9, Psalm 127:3-5, Luke 1:36, passim). Especially Ex 23:26 underlines my point: God promises a place that knows no infertility.

This is not contradicted by Matthew 22. Jesus speaks of marriage, not procreation. In Paradise there was no marriage either yet procreation was intented to exist there.

Additionally: since there was no pain in paradise when giving birth (cf. Gen 3 where painful childbirth is mentioned as a consequence of the Fall) there neither will be in the New Creation, so it will not do any damage to women to bear many children.

Also, there is no mentioning that the space will be limited.

After all, this question is an adiaphoron, though an interesting one.

As @Matthew's answer aptly presented, Jesus was asked this very same question. His answer was simple:

For in the resurrection they neither marry, nor are given in marriage, but are as the angels of God in heaven. (Matthew 22:30, KJV)

In the Bible, God's angels never come as male or female. They are not husbands or wives, and there are no angel babies. Angels do not marry, nor have children. Jesus is saying that in our transfigured, post-resurrection bodies, we will be like the angels in this respect.

There is a reason for this.

Throughout the sin-experiment on this earth, those who learn to rise above sin and who have obtained victory over it will be so inoculated against sin that they will never consider sinning again. As the prophet Nahum tells us:

What do ye imagine against the LORD? he will make an utter end: affliction shall not rise up the second time. (Nahum 1:9, KJV)

Sin will never again exist.

But what would happen if we continued to have children in the New Earth? Being told that sin is bad, and knowing this from personal experience, are two vastly different things. Sin could be experimented again, by one of those who did not know any better. And, given an eternity of time, it would almost be a matter not of "if" but of "when." Sooner or later, sin would again emerge. But this must never, never happen. Therefore, the creation of "free moral agents" ceases after this sin experiment is finished.

Most who try to say marriage will continue in Heaven do so because they are addicted to the physical aspects of marriage and cannot bear the thought of giving these up. However, we are told in the Bible that the wonders of Heaven surpass our imagination. We will not even miss anything that we no longer have. It will be like discarding the worthless in favor of something of greater value than we have ever known.

But as it is written, Eye hath not seen, nor ear heard, neither have entered into the heart of man, the things which God hath prepared for them that love him. (1 Corinthians 2:9, KJV)

God knows what will make us happy, and is prepared to do so beyond our wildest imagination.

Not all Christians follow Jesus' words here to the same conclusion, but the belief that there is no marriage in the New Creation¹ is common.

One article notes that there will likely be no need for marriage, as we will be with God and will have abundant companionship, and also that procreation will likely have ceased (after all, infinite time, zero death rate, and non-zero birth rate is a recipe for a population problem). Others note (1, 2) that marriage is an earthly shadow of the relation between Christ and the Church. Since that relation will be fully realized in the New Creation, it seems likely that its lesser echo will no longer be needed.

That said, there are dissenting beliefs. Personally, I tend to feel that such beliefs fail to resolve the very Question asked (as well as raising others, such as how it can be fair for someone who died unmarried to be forever less than those that did marry), and that the dissolution of marriage at the Resurrection seems more satisfactory.

(¹ Despite popular use, "heaven" is a misnomer and does not refer to the Christian's promised reward or the place redeemed Christians will inhabit.)