As @mikenichols points out, a big problem is going to be the passing on of harmful genetic mutations through the forced incest.
If they are so unlucky that they both carry the same defective gene, this could be devastating. 25% of their children would display the defect. Combined with other genetic defects, this could be brutal. But if they're reasonably lucky and they don't, their grandchildren will still have a high infant mortality rate -- see Mike Nichols post and my comments on it -- but enough grandchildren should survive to carry on the human race. And the deaths will slowly reduce the incidence of the defective gene.
Presumably they will have less genetic diversity than the human race as a whole does now. While they'll be carrying many recessive genes, they're not going to have the full set of all the genes that are presently out there. (Barring some extraordinary coincidence, the odds against which would have to be astounding.) The best chance would be it the two people are genetically as far apart as possible, like if one of them is a white person from Norway and the other is a black person from Nigeria or some such. Still, plenty of isolated communities manage to survive and thrive. There have been tribes on remote islands or deep in the jungle who had no contact with the outside world for centuries, and yet seemed reasonably healthy. There is no reason to believe that this of itself would be an insurmountable problem. It may be that they just wouldn't have the genes to survive in some climates, or to live off of certain foods, etc, but that wouldn't be fatal.
Again referring to my comments on Nichols post, suppose it's true that the average person carries 60 mutations. Most of these are probably trivial: a funny shaped ear, a mole, that sort of thing. Some will be serious. It is likely that some will be serious enough to be debilitating or fatal in the right (or wrong) circumstances. Some that are not fatal themselves will prove to be fatal in combination with others.
Once lost, there's no way to get genetic diversity back. Well, mutations will create genetic diversity, but not the kind you want. Mutations are random damage, and random damage does not make things better. Try turning your Honda into a Rolls by throwing it over a cliff a few times and see how that helps. No one has ever observed an unquestionably beneficial mutation. Even if you buy that crazy evolution theory, to fit the observed facts you have to concede that beneficial mutations are extremely rare, so it would take hundreds of thousands of years to accumulate even a handful.
Another big problem that comes to mind: How do you maintain any sort of civilization or economy with such a small population? Suppose each woman has 10 children who survive to adulthood. That seems very optimistic, but even then: second generation = 10, third generation = 50, fourth generation = 250. Sustaining our present technology requires many thousands of specialists. You rely on the work and knowledge of others every day. Suppose they decide that they want to make some electrical wiring. They'd have to know where to go to find copper ore and how to recognize it. Then they have to know how to smelt it into usable metal and string it into wires. Will they make the insulation out of rubber? What does a rubber tree look like, and what is the process for extracting the rubber and turning it into a usable form? Etc. Even assuming that libraries survive and they can look stuff up, there are limits to how much one person can learn. They might be able to continue to use many existing artifacts, but building new ones or even repairing the ones they have would be very difficult. And many would rust or decay over time, so that within a few generations they're probably wouldn't be much of the old technology still usable. They'd have to start over in many ways. They'd have the advantage that if they're smart enough to keep the old books, than they wouldn't have to re-discover or re-invent things. This would be like the fall of Rome and knowledge preserved by monks times a million.
RE religious implications: The immediate descendants of Adam and Eve had to reproduce by incest, as did the immediate descendants of Noah. It wouldn't take any great theological leap to say, This is an extreme situation, God will surely allow us to break a general law as the alternative is the extinction of the human race, there is no harm to anyone other than ourselves, and there is more harm by not doing it than by doing it. I might note that personally, I'd see this as very different from a situation where the only way to survive is by harming others. I suppose there could be Christians or Jews or Muslims or whatever religion you suppose these people are who would dogmatically say that all commands from God must be understood as no exceptions under any circumstances, and therefore we have no choice but to allow the human race to end.
Addendum: Rate of defects
Confer Mike Nichols post.
Consider just one gene. Let's call the "good" gene "A" and the bad, mutated gene "a". Then let's assume that one parent is AA and the other is Aa.
Generation 0: 1 AA + 1 Aa
Generation 1: 50% AA + 50% Aa
Generation 2: 25% will be children of AA+AA, 50% of AA+Aa, and 25% of Aa+Aa
Of the AA+AA, 100% are AA
Of the AA+Aa, 50% are AA and 50% Aa
Of the Aa+Aa, 25% are AA, 50% Aa, and 25% aa
Summing this up gives:
AA=25% x 100% + 50% x 50% + 25% x 25% = 25% + 25% + 6.25% = 56.25%
Aa=25% x 0% + 50% x 50% + 25% x 50% = 0 + 25% + 12.5% = 37.5%
aa=25% x 0% + 50% x 0% + 25% x 25% = 0 + 0 + 6.25%
So the grandchildren will have 6.25% incidence of a pair of bad genes, and thus show the recessive characteristic.
Assuming that each parent has 2 such bad genes, with no overlap of the 2, then there are 4 total bad genes, each of which will show up in 6.25% of the grandchildren. The probability of a child not having any one of these is 100 - 6.25 = 93.75. The probability of having none of the four is thus 93.75 ^ 4 = 77.25%. So -- assuming we are talking about fatal mutations here -- about 23% of the grandchildren will have at least one of the 4 fatal mutations, and die.