When I think of dramatic question, it usually has to do with information that an audience has that a character (or characters) onstage does not have. This is the way that a playwright creates suspense and how an audience stays "engaged" in wondering what will happen next in the play.
From this perspective, an important dramatic question in the play is: Will Othello ((or for that matter, any of the characters in the play) figure out that Iago is up to no good? Iago does what most all of Shakespeare's villains do, he begins the play by laying out for the audience his plan, and then the suspense is created for the audience in waiting to see how far he will be able to go, and if anyone will discover or expose his scheme.
In Act I, Iago lays the groundwork for all his duplicity by saying:
Were I the Moor, I...
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