I'll start off by saying that it's great that we can have a thead as frank as this on here. There are some sites where it would be a recipe for disater to talk about sexuality.
*turns sheet over* now on to my opinion. I have absolutely no problem with the idea of a LGBT Royal, be they The Sovereign, The Heir, or a peripheral member of the Family. Sure no direct heirs can be produced, but The Throne or the Peerage can easily go to a Niece or Nephew, as many on here have already said.
As for whether or not such a Royal would be accepted, I could see the public being accepting of a LGBT peripheral Royal, in a more progressive country, but I'm not so sure about how accepting they would be of a LGBT Heir or Monarch. I might be wrong, though, and I hope I am. I can't see any of the associated Churches* wanting anything to do with a homosexual relationship. The Church of England wouldn't even give Charles and Camilla a Wedding as opposed to just a Blessing. This would unfortunately make it very difficult for an openly gay Royal to keep their place in the line of succession, but again I might be wrong. Plenty of Monarchies have made gay Marraige (or at least Civil Partnerships) legal.
Despite my doubts, I do hope that Royalty takes this next step. This is the twenty-first century, now. Why should being gay be any different to being infertile, or opting not to marry? (not asking anyone on here, just making the point) Sexuality is just one part of a person (obvious but true) and it's pathetic that marrying someone of the opposite sex is ideal, while marrying someone of the same sex (or at least declaring you are attracted to members of the same sex) will almost certanly cost you your place in the line of succession. Noone can stop a Royal from being gay. They can only be stripped of certain rights if they admit it to themself, their Family, and their country. The dinosaurs who would be mortfied if anyone in any Royal line of succession came out as gay, lesbian, bisexual or transgendered, are probably tolerating a LGBT Royal, without even knowing it. History shows us that a Royal can be gay without calling up alot of media outlets, standing on the Palace rooftop, and shouting "It makes me physically ill to look at the genitallia of the opposite gender!" There have been alot of LGBT Royals (as the Wikipedia list shows)and there will continue to be. There's just never been a Royal who has come out, and been treated no differently for it.
On another note, would it be too contentious or off topic to bring up the possibility of an intersex baby being born into a Royal Family? Even in Countries with equal primogeniture, how would a baby with both Male and Female genitals be titled? Would the baby be given an operation? (pretending it isn't required for survival) It's an unlikely scenario, but not not an impossible one.
Ramble over
*Emphasis on associated. I'm not lumping all religion together.