Iago. What say’st thou, noble heart?
Rod. What will I do, thinkest thou?
Iago. Why, go to bed and sleep.
Rod. I will incontinently drown myself.
Iago. Well, if thou dost, I shall never love thee after it. Why, thou silly gentleman!
My gut feeling is that Othello isn’t a racist play in the way that The Merchant of Venice is racist or The Taming of the Shrew is sexist, but my gut feeling in this case is not necessarily to be trusted, as Othello, unlike the latter two, is a very good play. I was trying to rationalise the gut feeling by saying that Othello is unique among Shakespeare’s major parts because he actually has a job, but then I remembered that so do Shylock and Antonio. The mechanicals in A Midsummer Night’s Dream and other subplot-dwellers don’t count, as they are figures of fun.
The debate over Othello’s skin colour, however, is pretty racist. The Wikipedia entry doesn’t go into the messy details but a good critical edition will contain enough exceprts from early-to-mid-20th century writers, especially the Americans, to give you the creeps. And playing him in blackface is racist and pretty stupid.