I think it's important to see that Othello goes beyond "race" per say. Though Othello, through Venetian convention is whitened, in the sense that he has given up his barbaric past to be a highly ranked soldier in a "Aristocratic" context, much of the racism centered around the fear of sex. Othello is very much impotent in this play, it is Desdemona that has the sexual appetite, a desire that will be used to blacken her in many ways. When her father states his case against the moor to the council, she makes allusion to her right to have her marriage consummated. The handkerchief she loses is a sign of the magic surrounding Othello has a moor, and a sign of virtue in his eyes. He cannot consummate his marriage, and as such, demonstrates faith through the red pigmented cloth. He refuses to stab her, again alluding to the fear, or rather incapacity of penetration, and "puts out the light", or rather, leaves Desdemona dead, or blackened, losing his whiteness as well in the end. Othello is therefore Black by nature, but White by nurture, until he murders his wife, destroying his white facade again. Do I think that Shakespeare was passing a social commentary of race? In the end, what Iago plays on to make the moor kill his wife, is toying with the "emotions" or "characteristics" of his "whiteness". The play states that he was not naturally jealous, that's a trait of nurture over nature, and as a high rank military in Venice, the nurturing is not 'his'. Though the Black man marrying a White woman can be viewed very much like today, the implications surrounding the tragedy aren't as simple as race.
Side note, Harold Bloom is a wonderful companion to any Shakespeare play. His book Shakespeare, the Invention of the Human is quite delightful. 
