The principal relationship destroyed in Othello is, of course, that of Othello and Desdemona. What is depicted first as a loving marriage deteriorates solely because of Iago's relentless assault upon Othello's trust in Desdemona, not through direct statements but the opposite: hints and innuendo. A typical instance of Iago's indirection is the following exchange:
IAGO: Did Michael Cassio, when you wooed my lady, know of your love?
OTHELLO: He did, from first to last. Why dost thou ask?
IAGO: But for a satisfaction of my thought;
No further harm.
OTHELLO: Why of thy thought, Iago?
IAGO: I did not think he had been acquainted with her.
OTHELLO: O, yes, and went between us very oft.
Iago is not saying anything, really. But obviously, if he came out with a direct accusation at this point (such as to say, "I think Desdemona was sleeping with Cassio"), Othello wouldn't have believed him and would also probably have told Iago to get lost. Later, he says,
I speak not yet of proof.
Look to your wife; observe her well with Cassio ....
I would not have your free and noble nature,
Out of self-bounty, be abused. (III, iii)
Iago's method is always based on a slimy kind of flattery. He says, but he does not say, amid constant protestations of loyalty to Othello and compliments to him, when in reality, he hates Othello. But it is not only Othello's marriage to Desdemona Iago is destroying. It is also, ironically in the extreme, his own relationship to Othello, whatever that has been before Iago has taken it into his head to bring Othello down. It was a friendship based on deception, since Iago only cares about himself, but it was still a friendship at least outwardly. The subtext of Iago's scheme may be a kind of jealousy of Othello's devotion to his wife. But his stated reason for destroying Othello is the preferment, as Iago tells Roderigo, of
one Michael Cassio, a Florentine ...
That never set a squadron in the field
Nor the division of a battle knows. (I, i)
Thus, Iago destroys the trust Othello has in Cassio by staging the fight in which Cassio wounds Montano. Again, Iago uses indirection and deception to cause Othello to think the fight was Cassio's fault:
IAGO: Touch me not so near
I had rather have this tongue cut from my mouth
Than it should do offense to Michael Cassio. (II, iii)
Last, Iago can be said to have destroyed his own relationship to Emilia by forcing her to take part in the plot. His relationship to the rest of the world is terminated as well at the end of the play, because the full story has come out and Iago has been arrested (and is apparently going to be tortured as well). But all he has cared about is to destroy Othello, and Iago has accomplished this after Othello murders Desdemona and then commits suicide.