Gay / trans wouldn’t be a problem. Plenty of Gay men have impregnated women to keep up appearances or due to family pressures. And if the flesh is weak then there’s always a Turkey baster.
A transsexual man would almost certainly be biologically incapable of impregnating a female, and if you actually meant a transsexual woman, if she’s gone through surgery and/or hormonal treatment, she might have become biologically incapable of impregnating a female.
We did have a poster once whose child was sired by her transwoman lover, leading to much derision after the phrase “woman sperm” was posted. As she was also (and I assume still is) married to a cisman (who happened to be sterile) the family dynamics were a bit complicated.
Which, I hasten to add, was only a small portion of that person’s contribution to this forum.
The OP basically asked whether problems with inbreeding would necessarily preclude a single pair of humans from repopulating the world. The answer to that is “no.” (As I mentioned, the present population of 250 of Black Robin of the Chatham Islands are all descended from a single fertile female. Their population size is mainly limited now by availability of suitable habitat. Otherwise they would be able to increase even more.) Obviously, however, the more individuals and more genetic diversity you have the better odds a population has for long term survival and increase.
Even if there are detrimental recessives present in a population, that does not necessarily doom it. If they are lethal, however, they may reduce the rate of increase. Insular groups like Mennonite and Hasidic Jews have detrimental recessives, but these do not prevent these groups from having a high rate of increase.
Other factors besides inbreeding can be a threat to very small populations, in particular chance deviations in sex ratio. If your founding pair had all boys then you’ll be out of luck, even in the absence of genetic problems.
The OP only asked about inbreeding, but of course there are also extrinsic threats to the survival of very small populations, such as natural disasters, weather events, etc.
For animals in their natural environment, a Minimum Viable Population can be calculated. This is usually calculated as the population size necessary to ensure between 90 and 95 percent probability of survival between 100 and 1,000 years into the future. Calculated MVPs are often in the range of 500-1000 when inbreeding is ignored, and several thousand when it is included.
However, this says nothing about what the minimum possible founder population might be, just the population size you want to get up and maintain to to ensure long term survival. Also, it refers to populations of wild animals in the absence of human intervention. Humans would be better able to protect themselves from the impact of natural disasters. They would also potentially be able to consciously manage inbreeding problems (as human societies already do with prohibitions against incest).
I’m really surprised at that, thank you everyone for a fascinating thread, this is why I use the Straightdope. 
We’re also glossing over another issue - reproduction ability. I know of one couple who tried hard and took 6 years to have 2 children, a few miscarriages, and an abortion when ultrasound showed twins with no brains developing. Another couple who had given up, until after almost 20 years of marriage she became pregnant (and only had one child). Another couple I know have never been able to conceive despite trying. (The wife joked “my aunt asked me about birth control, I laughed and told her I haven’t had to worry about it for the last 10 years”)
So we are assuming our founding couple(s) can pump out kids one after another. Not every woman can do this. Like genetic problems - luck of the draw.
I’m ready to try a survivor’s experiment with the Victoria’s Secret models, even if they all have Victoria’s Hemophilia. It’s just the kind of giver I am.
menagerie a trois?
The examples quoted above for population recovery are for intensively managed species. The point is that birds and animals aren’t programmed specifically for “species recovery”. If you have 2, they may just not like each other. One or both may be LGBTIQA+. Some may be too old for breeding. If they have distinct ranges, they may never meet.
If the aliens manage species recovery, they may one male human in China, and a couple of young women in Aus. They may still have difficulty convincing their specimans to mate. If their aren’t any aliens, the man in China and the Australian women will never have children at all.
Furthermore, without a functional society, they are all going to find it very difficult to get enough to eat: some of the early European/American settlers would have starved to death if not fed by the local population.
If we take the hopefull postion that the breeding pairs are from an existing social group in a subsistance culture, like a couple of villages in PNG, then genetic diversity may be a problem. If you’re talking about NYC, then maybe they’d all be dead before that.
If the transporter doesn’t work.
Eh, if the Last Couple on Earth are in NYC, they probably won’t have a hard time for food. The city’s grocery stores have about a week’s worth of food for 8 million inhabitants. Even if half of that is perishable, that’s still 80,000 people-years worth of food.
Given the half-life of a twinkie, they are probably safe for a few hundred years!
The best really long-term example I know of is Tasmania (several thousand people cut off for about 10,000 years).
True, but that’s the debate - isn’t it? Most likely Tasmania was at the carrying capacity of the land before it got cut off; all that population did was survive, maybe dwindle as the climate changed and the land area grew smaller?
Whereas with Easter Island, a single flotilla, maybe 50 to 100 people, two canoes(?) created a population eventually guessed to be somewhere between 3,000 to 15,000 or more.
The number responsible for populating the Americas initially may have been several thousand or less before rising waters separated America from Asia. The population before the European diseases arrived has been estimated to number tens of millions.
But the point with either example is that a small population can continue, expand if resources allow, and rarely does it appear that less diverse genetics are a problem. Once the first few generations have ensured that there will be sufficient numbers to carry on, things seem to go well.
I suppose I should also mention Pitcairn Island, where…
By the time the outside world came back and the sole Bounty survivor was an old man, there was a healthy population on the island. It should be noted though that an earlier settlement by Polynesians failed because the island(s) of the Pitcairn group did not have the resources to support a full Polynesian community.
Of course, that included some other immigrants, especially missionaries.