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 would not be Iago:
In following him, I follow but myself.
. . .I am not what I am.
And, once Iago has promised the audience that he is going to act in a false way, the question kicks in. So, for a dramatic question that involves an actual audience participating in a performance of the play, I think the suspense of "waiting to see" whether Iago will be revealed as the villain he is the major dramatic question of the play.
This is a very intense question. I think that the dramatic question of the play revolves around the main character. One of the saddest elements I see in Othello is the constant faith he has in his own insecurities and ability to doubt. Othello strikes me as a character who is more predisposed to be miserable because he believes in human goodness, but cannot act on this belief. In this light, the dramatic question I have is whether he is going to be able to believe in the goodness of Cassio or Desdemona. Iago is able to work his manipulation to such a perfect degree onto Othello because the latter is so willing to see the worst in people. This is quite interesting to note because Othello is, for all practical purposes, an honorable man and one who believes in honor. While he defends this on a political level in both his actions as a military man as well as his position as a leader of others, he cannot accept this in his own personal setting. He is a being who understands human decency, but cannot accept it when it is the closest to him. This might be why the work has so much relevance on a personal and intimate level for its dramatic question drives to the heart of all who witness it.