What clues do we have in the dialogue that the cultural setting of Othello includes racial tension? 

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A lot of the racial slurs spoken by characters in Othello occur in the first scene of the play.  The characters of Iago and Roderigo are on their way to alert Brabantio to the elopement of his daughter Desdemona with Othello, the Moor.  Though Iago is explaining the situation in which Othello has promoted Cassio over him, he never calls Othello by his name, but rather by his race -- The Moor.  This is a common racial slur, indicating that the speaker does not think of the person about whom he speaks as an individual, but rather as a part of group of people, a stereotype.

Roderigo, though he doesn't have much to say, also manages to refer to Othello with a racial slur.  At line 66, he calls Othello "thicklips."

Once they arrive at Brabantio's, Iago hides in the shadows and does not give his name, but becomes even more crude in his references to Othello.  He calls him a "Barbary [Arab] horse," says that Othello and Desdemona are "making the beast with two backs," and refers to Othello as "an old black ram."  He even refers to Othello as "the devil."  The sum of these slurs is that Othello is an animal and less than human.

Roderigo picks up the term "Moor" in his reference to Othello and says, at line 126, that Desdemona is in "the gross clasp of a lascivious Moor."

All of these slurs occur in Act I, scene i.   And they all occur before Othello even enters.  And once he does, he is revealed to the audience to be noble, dignified and much respected.  In this way, Shakespeare reverses the expectation of the audience (and maybe some of their prejudice about Moors as well).  By having all the slander and racial comments in the opening scene, Shakespeare sets the audience up to expect a comic, inept character.  Instead. they meet one of the most eloquent speakers in all of Shakespeare's plays.

For more about race and Act I, scene i, please follow the links below.