Shakespeare wrote Othello during the European Renaissance, and it certainly deals with racial issues. However, the only people who express or seem to feel racial prejudice toward Othello are Iago and the people he influences. The dignitaries of Venice respect him and defend his right to be married to Desdemona. So the play reflects an increasingly cosmopolitan European society at the time of the Renaissance.
As for the Harlem Renaissance, it represents an emergence of African-American contributions to the arts. If Paul Robeson, who acted the role of Othello, is considered part of that movement, then there's a connection, but you may be able to find a stronger one.