Iago's suspicion is just an added weapon in his acidic arsenal for revenge against both Othello and Cassio. He hates the two so much that he continuously seeks reasons to support his bitterness and lust for revenge. His resentment for the two stems from the fact that, first, the general humiliated him by not appointing him as his lieutenant, a position Iago felt he was entitled to since he had been so loyal to Othello. And, second, that Othello named the young and inexperienced Cassio, a foreigner, instead. Furthermore, Iago seems to have always disliked the idea of Othello, also a foreigner, being his master. Also, he despises the fact that Cassio is young and handsome; qualities he apparently does not have.
Believing that both men had had sexual relations with his wife, Emilia, strengthens Iago's resolve to punish them for their so-called indiscretion. He then ruthlessly proceeds to attempt the destruction of both men—a venture in which he achieves both success and failure.
He manages to manipulate Othello and Cassio to such an extent that Othello later dismisses Cassio. Othello is also so overwhelmed by jealousy due to Iago's sly orchestration that he kills Desdemona, his beautiful wife, and later commits suicide. Iago's bid to have Cassio murdered fails when his unfortunate puppet, Roderigo, does not kill him but is himself mortally wounded in a planned skirmish with Cassio.
Iago, in a soliloquy, first declares his suspicion that Othello has had an affair with Emilia at the end of act 1, scene 3, where he states:
I hate the Moor:
And it is thought abroad, that 'twixt my sheets
He has done my office:
By saying that Othello "has done my office", he means that Othello was doing the duties that he, as Emilia's husband, is supposed to perform: having carnal relations with her. However, Iago does admit that he is not quite sure whether he should or should not believe the rumors but that
I, for mere suspicion in that kind,
Will do as if for surety.
He will, therefore, conclude that the stories themselves are substantive proof for why he should hate the general. In his warped thinking, Iago clearly believes that he is defending his honor.
Iago again, in another soliloquy, confirms his belief in act 2, scene 1, when he says:
For that I do suspect the lusty Moor
Hath leap'd into my seat; the thought whereof
Doth, like a poisonous mineral, gnaw my inwards;
And nothing can or shall content my soul
Till I am even'd with him,
In this instance, he makes his intentions obvious. He wants to get even with the general for cheating with his wife. "Hath leap'd into my seat" as in the previous instance, makes it clear that he believes that Othello had usurped his position and committed adultery with Emilia.
In the same soliloquy, he also states his suspicion of Cassio:
For I fear Cassio with my night-cap too—
He suggests that Cassio might have also slept with Emilia and that it is, therefore, his duty to punish the lieutenant for his transgression by making Othello believe that his trusted lieutenant was doing the same with his wife, the beautiful Desdemona.