Why do you suppose that Shakespeare does not have Iago die in the final scene of the play?
To kill Iago would be to relieve him of the guilt and misery he experiences. Although it is hard to believe Iago has any feelings at all, his main disappointment is that he has lost all chances of becoming a high-ranking official. Iago would rather die than live with the knowledge that all his scheming came to naught.
In the play, dying is seen as an honorable and even comfortable end. There are several examples of this. One is the fact that Othello kills Desdemona in their marriage bed. Emilia stands her ground against a crazed Othello, accepting her fate and welcoming the chance to die next to her "lady" Desdemona. Another example is Cassio's handing the dagger to Othello; it is Cassio's final act of respect and honor to Othello, allowing Othello to depart on his own terms without having to endure the humiliation and disgrace of being imprisoned.
You may often hear or see Iago being referred to as Othello's "ensign." An ensign is sometimes also referred to as a "standard-bearer." In either case, Iago serves as a military officer, "a man of...of loyalty and trust" to Othello's face, but he turns out to be a villain and an "inhuman dog" who is egotistical and evil.
Iago may not die onstage, but his doom is certain. In the play's final lines (5.2.377-381), Lodovico directs Cassio, in his first assignment as the lord governor:
...To you, Lord Governor
Remains the censure of this hellish villain.
This time, the place, the torture, O, enforce it!
Myself will straight aboard, and to the state
This heavy act with heart relate.