Is there a consensus on whether or not race exists on a biological level?
Anatomically modern humans are less than 200K years old, left Africa (or possibly Africa and Arabia) only 60K years ago, and migrate quite a lot (think of the Mongol invasion).  So you would not expect to have clear divisions into subspecies.  And indeed a quick search of google scholar pulls up no claims in the literature for human subspecies of anatomically modern humans (i.e. no subspecies of homo sapiens since Neanderthals and Denisovans).

On the other hand, it's certainly the case that you can work out a rough phylogenetic tree of humans.  Native americans are descended from certain northern and eastern asians, and so in the 1400s were more closely related to them then they were to other humans.  Similarly, Austronesians (of Indonesia and Polynesia, e.g.) are more closely related to Taiwanese aborigines than they are to Australian aborigines.  This tree is complicated by how much more migration there has been in the last 1000 years than there was previous to that, but it's still biologically meaningful.

But here's the striking thing: traditional notions of "race" do not match up well with this tree.  The deepest and most fundamental phylogenetic splits in the human tree are within Africans.  If you were going to try to "scientifically" come up with races, you'd end up with something vaguely like one for Khoisan, one for other Africans, and one for all non-Africans.  Even ignoring modern admixture, traditional races are not monophyletic.  Any clade including all "black" people includes all people, any clade including all "east asians" includes all native americans.  (Also note that traditional notions of "race" are not stable over time, remember that "Irish" used to be a "race" on the US census.)

Modern admixture even more clearly shows the non-genetic nature of traditional "races."  One genetic study of people who identify as African-American, found that African ancestry can be as low as 1% and as high as 99%, with the median African-American having 18.5% European ancestry (with 30% having European y-dna).  Furthermore, relatively ancient admixture (for example, in the Northern Africa, Northeastern Africa, Middle East region or in Madagascar) also complicates the situation significantly.

To summarize, although human genetic diversity is a real biological phenomenon which can be used to figure out the geographical history of people's ancestors, traditional notions of race do not match up well with the biologically meaningful concepts.