Occasionally, the Old Testament (OT) prohibitions on same-sex sexual activity are used as part of a Christian theological case for the impermissibility of same-sex sexual activity. There is one weakness in this argument that I want to address (I do not want to deny that there are other issues), and this is that some OT rules--most evidently dietary ones--have been sublated in New Testament times and are binding on the Christian in a non-literal form (e.g., the prohibitions on some foods that come into a person's mouth might be transformed into a prohibition on the speech that comes out of a person's mouth). The traditional answer is to distinguish between "ceremonial" and "moral" precepts, and to claim that the ceremonial ones no longer literally bind those who have died in Christ (though they may exist in a sublated form), but the moral ones are eternal. The opponent of the argument against same-sex sexual relations may well claim that the prohibition in question falls in the ceremonial category.
But I think there is a different way of fixing up the OT-based argument for impermissibility. I shall claim that it would be inappropriate for the God of love to prohibit same-sex sexual relations unless these relations are wrong. Here is the argument.
After all, there are persons whose sexual attraction is exclusively towards members of their own sex. Some highly motivated such individuals do apparently succeed in changing their attractions to opposite-sex ones, but it does not seem that the majority succeed (I've heard from a colleague that the best data indicates that about 30% of highly motivated same-sex attracted individuals can change to have an attraction for the opposite-sex). It is also plausible that same-sex attracted individuals unable to change to being opposite-sex attracted existed in not insignificant numbers in ancient Israel. The defender of same-sex sexual relations is unlikely to deny this.
Now, flourishing in a morally upright romantic relationship is one of the central parts of human flourishing (that does not deny that some might appropriately sacrifice this form of flourishing for the supernatural goals of celibacy), and it is particularly a flourishing in respect of our capacity to love. If same-sex sexual relations are morally permissible (apart from divine prohibitions), then same-sex romantic relationships will, surely, be a central part of the potential human flourishing of same-sex attracted individuals. Furthermore, sexual relations within marital commitment are the consummation of a romantic relationship. If same-sex sexual relations are morally permissible, then to prohibit same-sex sexual relations to an individual incapable of opposite-sex sexual relations is to prohibit the individual from exercising a central part of human flourishing. And this seems an inappropriate thing for the God who is Love to do, and whose purpose for us is to fulfill our love. (In fact, I think a number of the statements in this paragraph are standard parts of the case for the permissibility of same-sex sexual relations.)
In summary, if same-sex sexual relations are morally permissible (apart from divine prohibitions), they enter into the consummation of morally upright romantic relationships for people incapable of flourishing within opposite-sex romantic relationships. But if so, then it was inappropriate for our God to have forbidden them in OT times. But God did forbid them in OT times, and God does not do what is inappropriate. Hence, it was appropriate for our God to forbid these relations, and hence the relations were not morally permissible. But morality itself does not change (though ceremonies do), and hence even now they are not morally permissible.
Observe that a similar argument cannot be made in the case of clearly ceremonial precepts. E.g., ham is yummy, but eating ham is not central to the fulfillment of human individuals. Even if one were biologically constituted so that ham is the food that would taste best to one, it would be false that there is a central part of human flourishing to which the eating of ham would be essential. So even though eating ham is not morally wrong (in itself), it would not be inappropriate for our God to prohibit it to a segment of the human population.
Objection 1: Perhaps God had made a special blanket call for a segment of the Israelite population (namely, the same-sex attracted population, or at least the portion of it unable to change the attraction) to engage in the supernatural self-sacrifice of chaste celibacy.
Responses: This seems implausible. First, widespread celibacy for a supernatural reason seems to be a new thing in Christian times. Second, there is no indication of a special supernatural goal being given to this segment of the population, though it is possible that one was given, but no data survives about this.
Objection 2: God has absolute authority over us. He would be fully within his rights to prohibit blue-eyed people from engaging in sexual relations with anybody. We have no right to our human flourishing--it is all a gift of God.
Response: I think there is much to this objection. Indeed, I think it is the most powerful objection to my argument. However, even though such a prohibition would probably be permissible to God in the abstract, it does not seem to fit with God's plans for the human race as shown in Scripture. God gives us life, life to the full.
Objection 3: The objection proves too much, because it also shows that God is not within his rights to allow impotence to happen (since impotence makes the marital consummation of romantic relationships impossible). But impotence happens, and God is omnipotent, so God must be allowing it to happen.
Response: There is a difference between permitting and doing. God permits evils to happen to us, in order that a greater good might be instantiated, but that is not the same as positively doing evil. However, if God actually prohibited same-sex sexual relations, and these relations were a crucial part of the human flourishing of some individuals, then this would be a much more direction relation between God and the evil--it would be like God forbade us from breathing. A prohibition is an action.
Objection 4: The commands in the OT are not the word of God but the word of humans.
Response: I said I would be responding to a specific objection to the OT-based argument against same-sex sexual relations, not against every objection. Objection 4 applies to just about every Scriptural argument. Those like me who accept the divine inspiration of Scripture (without denying that the human authors were authors, along with God) will not find Objection 4 particularly compelling.