Is it really detrimental to humans as a species to reproduce with one's own first cousin?
Inbreeding depression, "the reduced survival and fertility of offspring of related individuals" (quoting the linked article), is a well-known and well-understood biological effect. It does, indeed, affect humans.

The problem is that recessive mutations become more likely to affect the survival of the offspring of relatives. Imagine that you have a mutation affecting, say, some gene that's critical to your metabolism, which is entirely possible. But, since you have two copies of the gene and only one of them is faulty, you are not affected by this. Nevertheless, you, by chance, pass the non-functional one on to a couple of your kids, who are also not affected due to only having one copy as well. They, in turn, pass it on to one or more of their kids. Now, two of them, cousins, have a kid. The kid gets both copies and can no longer properly metabolize something due to having two faulty copies of the gene. This can prove to be fatal or at least severely debilitating.

Now, lest you think this is improbable, there are on average 70-something mutations per generation. Some of those will be neutral in effect, others will be detrimental. Once an offspring gets two copies of those detrimental alleles, then there could be serious health effects. The likelihood of an offspring getting two copies decreases as the relatedness of its parents decreases. So, sibling or parent/child pairings would be the most likely to produce offspring with the problems. Pairing between 3rd generation (First-cousins) or any pairings between them and their siblings, parents, aunts/uncles or grand-parents would be the next most likely.

So yes, first-cousins marrying and having kids is a bad idea for very basic scientific reasons. 



I missed the "as a species" part of your question. Inbreeding will only likely have an effect within small, closed populations, though it will continue to have a lasting effect even as those populations grow and open up. Two textbook examples are French Canadians and Ashkenazi Jews. Even now, there is continued elevated risk of certain rare genetic diseases in these populations. As for the species as a whole, it is likely to be really detrimental only if the effective population size of the species becomes really low: a general rule of thumb used by conservation biologists is that the effective population size should be at least 50 to avoid the effects of inbreeding (and 500 to avoid the effects of genetic drift) (50/500 rule).

Sorry about the confusion.