This is a passage that has been tragically misrepresented for millennia, causing intense confusion (to say nothing of immeasurable sadness).
Getting married vs. Being married
The two key verbs are γαμέω (gameo, to marry) & γαμίσκω (gamisko, to give in marriage). These verbs describe the act of getting married, not the state of being married. (In English we could also articulate this difference by saying I got married on X date versus I have been married for Y years).
Marriage is an ordinance carried out on this earth. This passage in Mark--and the parallel passages in Matthew & Luke--indicate that people do not get married in eternity. They say nothing at all against the state of already being married.
To claim from these verses that people in heaven cannot continue to be married to someone they married in this life is grammatically incorrect; this is not what the passage says.
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The Levirate Marriage
The principal reason many have interpreted these verses to rule out the possibility of being married in heaven (contrary to the grammar of the sentence) is the statement (in Matt & Mark) that they are like angels in heaven.
Luke gives more detail and offers greater clarity:
36 Neither can they die any more: for they are equal unto the angels; and are the children of God, being the children of the resurrection.
Some conclude that Luke contradicts Matt & Mark here. While that is one possible interpretation, it certainly is not the only one.
The question that sparked this discussion was the hypothetical in which a married man dies, and 6 of his brothers in succession marry his wife (in Levirate marriage). The Levirate marriage is a temporal arrangement only--its purpose is to address temporal matters of this life (including raising up seed)--it has no binding standing in the next life. The 6 brothers who participated in Levirate marriage were not married to the woman for eternity; they will be (presumably) single in eternity and they will be like the angels in heaven (some refer to this as the role of "ministering angels").
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Marriage in God's plan--a relationship that does not end
As already noted by NigelJ, marriage is frequently used as a metaphor for people's relationship with Christ in eternity--a relationship that never ends.
Jesus Himself taught of marriage:
Wherefore they are no more twain, but one flesh. What therefore God hath joined together, let not man put asunder. (Matt. 19:6)
God has made them one. They are no more two. There is nothing in scripture that reverses this, God's mathematical axiom of marriage.
Additionally, if marriage were for this life only, any time a married person died at the hand of another person (on purpose or by accident), the perpetrator would be putting asunder what God had joined together, permanently ending a marriage relationship. But it gets worse. Since by man came death (1 Cor. 15:21, speaking of the Fall of Adam), all marriages ending with death are being put asunder by man. Since death is the fate of all mortals, Jesus would effectively be saying "What therefore God hath joined together, let not man put asunder...oh and by the way, man does put asunder all which God hath joined together."
The idea that man is forbidden to break a relationship between man & woman put together by God is meaningless if man does in fact break apart every relationship between man & woman put together by God.
Let's turn to Paul.
The last enemy that shall be destroyed is death. (1 Cor. 15:26)
Death is swallowed up in victory. O death, where is thy sting? O grave, where is thy victory? (1 Cor. 15:54-55)
If death does permanently rob people of the relationship--given by God--that more profoundly effects them, motivates them, and matters to them, than any other mortal relationship...death does take from people something the resurrection cannot restore. That's a pretty nasty victory for the grave.
Finally, Paul's most theologically profound statement on the subject:
Nevertheless neither is the man without the woman, neither the woman without the man, in the Lord. (1 Cor. 11:11)
Alone, we cannot be what we have the potential to be. We cannot be what God intends us to be.
The marriage relationship created by God is intended to endure beyond the grasp of death, and is far more than just a convenient means of producing babies. To reduce marriage to making babies and nothing else is to commit the very error Jesus reprimanded the Sadducees for making.
It is a Levirate marriage whose explicit purpose is temporal in nature. The instructions for Levirate marriage (see Deut. 25:5-10) acknowledge the original husband's relationship is superior/transcendent to the brothers' Levirate marriages. The first brother's marriage is not a Levirate marriage, it is not equivalent to brothers 2-7's marriages to the woman.
To put it in a syllogism, if the Levirate marriage is about raising up seed & managing temporal concerns, and the original marriage was more than that, then the original marriage was about more than just raising up seed & managing temporal concerns.
This concept is found in Genesis as well. When God brought Eve to Adam it was to be a help-meet to him:
And the Lord God said, It is not good that the man should be alone; I will make him ban help meet for him (Genesis 2:18)
The command to procreate is given in Genesis, but Genesis clearly rejects the misogynstic view that the purpose of women is simply making babies. God made men & women as help-meets to support one another and to supplement each other's abilities. This is a purpose that is more than just the physical vicissitudes of Levirate marriage.
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Gender
So God created man in his own image, in the image of God created he him; male and female created he them. (Genesis 1:27)
God created humans male & female. He did so before the Fall. He did so before giving the commandment to procreate. There were male and female in this paradisiacal state. There will be male & female, and there will be children, in the paradisiacal state to come (see Isaiah 11:8, Isaiah 49:22-23, Isaiah 52:1-2, Isaiah 54:13, Zeph. 3:14, Zech. 2:10)
The idea that there would be no females in heaven is a Gnostic heresy that appears in the 2nd century (see Gospel of Thomas 114); it is a concept entirely foreign to the Bible.
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Conclusion
Does the verse above imply that those who attain eternal salvation will lose their gender and become gender-neutral angelic beings at the resurrection? No, this is a variation on a 2nd century Gnostic heresy.
If so, why were humans not created gender-neutral from the very beginning? N/A, see previous answer. Gender pre-dates this fallen world and will post-date it as well.
Was sexual reproduction designed to last for a period of time and then disappear? The passage does not address this topic.
Getting married does indeed appear to be an ordinance reserved for this life. Being married is a state God intends to endure for eternity. If people follow God's intended course, the man is not without the woman and the woman is not without the man. They were made for each other. Neither was designed solely for the purpose of procreation.
Happy Fathers Day.
