If we did a "Noah" project today, what would happen genetically?

Oh I think there would be all sorts of problems! There would be a higher risk of genetic disorders. Also I think we would end up losing species because you couldn’t guarantee that all the animals would be able to successfully reproduce.

This is a good question, I’m looking forward to seeing what everyone else has to say.

We can get a glimpse of what would happen by looking at species that have experienced a “population bottleneck” at some point in the past and had to regenerate from a radically reduced gene pool.

The Cheatah is one such example. Environmental factors appear to have eliminated all but one species of cheatah, and decimated that one, about 10,000 years ago. The result is that, even with whatever genetic mutations have occured since then, all modern cheatahs are almost genetically identical. This has caused huge problems for the species, including poor sperm quality and greater susceptibility to disease. If, for instance, a given virus is deadly for one individual, chances are great that it will be deadly for all individuals. Scientists are working hard to save the cheatah by actively encouraging what little genetic variance there is in that troubled species.(source)

What @Harp said. The population bottleneck would lead to genetic drift, which is basically an accident of probability wherein traits get passed on to offspring purely due to random chance, whether they’re beneficial to the species or not. (For instance, if something has a 50% chance of being transmitted, in every new generation it probably won’t be passed on exactly 50% of the time—it might show up in 54% of the next generation. So you might suddenly see a trait increase in frequency even though it confers no adaptive advantage.)

In large populations, natural selection acts against this by killing off the offspring who received maladaptive traits and not killing off the ones who got good traits. In a population bottleneck, however, the effects of random chance (on survival and subsequent DNA transmission) are much more pronounced because arbitrary transmission of a given trait isn’t diluted over a large population, and the effects of chance and luck—rather than fitness—become much more important in regard to what DNA gets passed on.

Over time, whatever DNA was passed on becomes “fixed” in the population—that is, every member has the same alleles at a particular locus. If this DNA was selected for by adaptive pressures, this tends to be a good thing and increases the overall fitness of the population. If the DNA was selected for by accident—which is what happens in genetic drift—it tends to be bad news bears for the population.

One reason this is bad is just because lousy traits have become more common in the population. Another reason this is bad is inbreeding depression. If you have studied a little bit of genetics, you know that people have pairs of alleles, and each of these alleles can be dominant (B) or recessive (b). You get one from each parent. If you get two copies of B (BB), the dominant trait is expressed. If you get B and B (Bb), the dominant trait is still (usually) expressed. You would only express the recessive trait if you get b from both parents (bb).

Sometimes maladaptive traits can survive in a population for a very long time simply because they’re recessive—that is, the gene is carried, but not expressed until you get two copies of it (bb). So if you have a population with increasingly similar DNA (due to drift), the population will start to get more and more copies of this harmful recessive allele, and it will be expressed more and more often, and the recessive disease will increase in frequency even though it’s killing off your population.

And that is why the Noah project would suck.

I think only rats and giraffes would survive.

Genetically, the human race from now on would look like Chuck Norris, because he would certainly survive and mate with whoever was left.