he had suffered during these times were what she wanted if any man wouldwoo her. Othello says he did just that—told her more stories.

OTHELLO:

And [she] bade me, if I had a friend that loved her,

I should but teach him how to tell my story,

And that would woo her. Upon this hint I spake:

She loved me for the dangers I had pass'd,

And I loved her that she did pity them. (175-180)

Othello loves Desdemona because of her gentle heart. However, it is her kind feelings that cause trouble because she takes pity on Cassio when he loses his position for drinking and fighting. Desdemona agrees to meet with him to see how she might help. (Iago uses the time these two spent together to poison Othello's mind against his wife.

Othello forgets his wife's love, consumed by an unfounded jealousy. For the pity she feels for him and others, he believes that she has been unfaithful, and he kills her. Othello also believes Iago's stories over his wife's words when Iago tells Othello that Desdemona has been unfaithful. Though Desdemona denies it all and swears her abiding love for Othello, his doubt overrules his love, and he kills her.

Othello realizes his mistake too late. The tales he told, Desdemona accepted without question. Her protestations of love he disregarded. The things Iago told him he accepted without question—and he doubted his wife.

The term "poetic justice" dictates that...

...in literature, good should be rewarded and evil punished. 

Many critics in the past saw that Shakespeare had not addressed "poetic justice" as he should have in the play. Based on this definition, we see that Desdemona is good, but is punished—as are Iago and Othello (who are both evil). There is no poetic justice in this sense.

However, please note that Percy Bysshe Shelley studied Othello and looked deeper to discover an accurate interpretation of the play. Shelley concluded...

...in Othello Shakespeare was portraying a man whose misfortunes were due to the intrigue of another, and were not intended by the dramatist to appear as retribution for any of his own misdeeds.

Shelley believes that Othello was not punished for being evil, but was victimized by evil Iago.

However, based on the earlier perceptions of Othello (and poetic justice), he was evil; "double poetic justice" is seen in his punishment because he believes the evil Iago, and disbelieves the innocent Desdemona. From an old-school perspective, Othello is punished in losing the woman he loves, and loses his life—because he is evil. 

Poetic justice appears in a variety of ways in Shakespeare’s Othello. Thus Othello dies after first having killed Desdemona; Emilia dies after having inadvertently contributed to Desdemona’s death; and Iago is badly wounded after first having badly wounded Cassio.

One particularly striking example of poetic justice appears when Othello, after having killed Desdemona, is relieved of his command in Cyprus. Lodovico, in the play’s final scene, says to Othello,

You must forsake this room, and go with us: 
Your power and your command is taken off, 
And Cassio rules in Cyprus.

These words inevitably recall the scene, much earlier in the play, in which Othello had stripped Cassio of authority:

Cassio, I love thee 
But never more be officer of mine.

Othello, as the play concludes, suffers a kind of double poetic justice: not only is he stripped of authority as he once stripped Cassio of authority, but the authority of which Othello is now deprived is in fact given to Cassio.  The poetic justice thus seems doubly emphasized.

There are two schools of thought with regard to "poetic justice" in Shakespeare's Othello.

Early on, "poetic justice" was a literary device, which required...

...in literature, [that] good should be rewarded and evil punished. 

Early critics of Shakespeare's works took this definition seriously, and found Shakespeare wanting.

The burden of the critics from Rymer to Johnson was that Shakespeare had violated all our fundamental notions of "poetic justice," or in other words had paid no attention whatsoever to moral considerations.

Shakespeare was austerely judged by some because he had not addressed these criteria in Othello: Desdemona is a loving and loyal wife. She should be rewarded based on this definition, but is punished. While Iago is unquestionably evil (and is punished), Othello is more a tragic than evil man. Even so, based on the definition, Othello is not good; so in killing his wife, he, too, is evil and should be punished. Because Shakespeare did not follow these guidelines, his work garnered some criticism.

However, Percy Bysshe Shelley, who was a fine Romantic writer in his own right, studied the works of Shakespeare, including Othello, looking for a clearer way to interpret the play. He did not see Othello as a villain at all:

...in Othello Shakespeare was portraying a man whose misfortunes were due to the intrigue of another, and were not intended by the dramatist to appear as retribution for any of his own misdeeds.

For those who find Othello an evil man on par with Iago, Othello's murder of his loving wife denotes evil, and therefore he should have received the same punishment as Iago for poetic justice to be served.

However, for those who are in agreement with Shelley, Othello is not punished because he is not evil. He is Iago's victim. However, he is still a tragic hero, and he does die (by committing suicide) at the story's end.