I would focus on the protagonists themselves. It might be a bit reflective in scope, but both strike me as very unsure of themselves through the eyes of the other. Both are highly sensitive to how others perceive them and their tragic conditions are highlighted by the lack of grounding in this regard. Simply put, their constant viewing of themselves through others' eyes helps to bring about their own destruction. Othello is driven to a negative degree by demons such as his status as a soldier, an outsider, a man of color, and his belief that his success is something that can be easily undermined. To a large extent, this helps him to accept Iago's machinations with so little resistance and to accept the belief that Desdemona is being unfaithful. It is in this same light that Gatsby carries himself with trying to win over Daisy. He lacks the perspective that would give him a sense of understanding which would indicate that his pursuit of her must stop at a logical and psychologically healthy end. Additionally, Gatsby is driven by his own background as one that must be concealed from a social setting that would reject someone like him. This compels him to spend extravagantly and carry himself in a manner without any sense of focus or regard for his own sense of self as he views himself through the eyes of "the other."
One starting place for a comparison between Othello and The Great Gatsby is the mutual theme of corruption. In Othello, Iago is the villain who corrupts perceptions of reality so that realities are hidden under a cloak of appearances. As Iago says of himself, "I am not what I am," he therefore embodies the corruption of reality and the resultant corruption of action, as Othello's actions prove.
In Gatsby, wealth and privilege corrupt behavior, which in turn corrupts reality in a reverse order of the corruption in Othello, in which reality is first corrupted then behavior follows. In Gatsby the wealthy believe they can--and do--act with impunity in ways that are devoid of any moral base of value or restraint.