Othello's most famous line is from Act V Scene ii, the final scene of the play: "It is the cause, it is the cause, my soul." Othello says this when he enters Desdemona's chamber to kill her and sees her sleeping. "The cause" refers to the crime of adultery, of which he believes her to be guilty, rather than to Desdemona herself--he does not want to kill Desdemona as vengeance, but as justice for her crime.
One of Iago's most famous lines is in the first scene of the play: "But I will wear my heart upon my sleeve/For daws to peck at." This line is often misinterpreted, because it is the second half of the thought. Iago's complete line means that he hides his feelings and intentions in order to protect himself. This is probably the origin of the modern expression "to wear one's heart on one's sleeve."
Iago has another line that has become a cliche: "O, beware, my lord, of jealousy!/It is the green-eyed monster which doth mock/The meat it feeds on," from Act III scene iii. Iago first plants the idea of Desdemona's unfaithfulness in Othello's mind, then warns him against being too suspicious, saying that it will make him miserable. This reverse psychology only serves to fuel Othello's jealousy further. The modern idiom "the green-eyed monster," meaning jealousy, comes from this line.
Othello's wife, Desdemona, has comparatively few famous lines. Her most famous quote is from Act I scene iii: "My noble father,/I do perceive here a divided duty." It is the beginning of a speech she gives to her father in which she says that her loyalty is split between him and her husband. She is explaining that, even though she is a dutiful and obedient daughter, she cannot do as her father wants because she owes even more obedience to Othello.