of Iago's hatred of Othello, "the moor," as Iago calls Othello, is not, making this a situation involving dramatic irony. 

If we take irony to be the discrepancy between appearance and reality, then certainly Iago is a character in whom irony abounds greatly in Act One. He takes great pains to appear to loyal confidante of Othello whilst actually working to act against him, informing Desdemona's father of Desdemona and Othello's elopement together. Note how he explains the way he must leave before Brabantio descends to Roderigo:

Farewell, for I must leave you.
It seems not meet, nor wholesome to my place,
To be produced, as, if I stay, I shall,
Against the Moor.

His whole identity is based on his hatred of "the Moor" and he takes great pains to appear to be loyal whilst plotting how to ruin his life. Note how in Act I scene 3, he says in a soliloquy:

He holds me well,
The better shall my purpose work on him.

Friendship and loyalty are just outward guises to allow him to advance his envious resentment against Othello.

From the commencement of the play, Othello is full of ironies, most of which are or become obvious to the audience through Iago's character.

"Honest, honest Iago," the least honest of all, steers the plot and is so manipulative that, as circumstances permit, he is able to confuse, convince and connive to the point of bringing out the worst in otherwise honorable characters.

The audience is not kept waiting long before Iago hints at his personality and warns - "I am not what I am" (I.i.65). The irony surrounding this comment is so intense as it is almost as if he is practising reverse psychology - after all he warned the audience right here at the beginning. Despite this, no one would expect such skill in maneuvering people to commit the most heinous deeds possibly because as Othello himself repeats many times  "Iago is most honest" (II.iii.7)

As most of the irony in Othello surrounds Iago, this supports the theme of appearance versus reality. Such is Othello's trust of Iago that he entrusts Desdemona to him. This act is however contradictory as, if he trusts Iago so fully and completely, why did he not make him his lieutenant?  

A man he is of honesty and trust. / To his conveyance I assign my wife" (I.iii.284-285).

This makes the circumstances doubly ironic as the plot of Othello could have been so different had Othello not angered Iago and, subsequently, without Iago's need to avenge himself and restore his (non-existent) honor and reputation, there would have been little or no story! 

Iago insists that reputation is laughable and tells Cassio so:

Reputation is an idle and most false imposition (II.iii.259)

but it is his feelings of rejection by Othello and the apparent tarnishing of his reputation that drives him. He is trying to convince himself as well as anyone else which in fact makes his whole character an ironic one. It is his own needs and his ability to manipulate situations that allow him to capitalize on the importance of reputation to any of the characters in Othello.

In Scene 3, Michael Cassio and Roderigo get into a fight and enter the scene where Montano intervenes and tries to stop the two. Michael, who is inebriated, then turns on Montano and starts a fight with him, injuring the governor. Roderigo has in the meantime run away and raised the alarm. Othello arrives at the scene and stops the altercation. He then asks Iago to explain the reason for the brawl. Iago recounts the following:

I do not know: friends all but now, even now,
In quarter, and in terms like bride and groom
Devesting them for bed; and then, but now--
As if some planet had unwitted men--
Swords out, and tilting one at other's breast,
In opposition bloody. I cannot speak
Any beginning to this peevish odds;
And would in action glorious I had lost
Those legs that brought me to a part of it!

Iago assumes a position of innocence and declares that the two men had just recently been on very friendly terms, getting ready for bed as a bride and a groom would do, peacefully. He furthermore states that their moods had undergone a very sudden change as if they had been affected by some planet that had made them act like idiots. They had drawn their swords and were involved in a bloody duel. Iago states that he does not know what could have possibly started the quarrel but that he wished that he had lost the legs that brought him to the scene for there would have been greater honour in not witnessing such a distasteful display.

This extract is thick in verbal irony, for it is Iago who had, with Roderigo, planned the brawl. He had previously told Roderigo that Desdemona was in love with Cassio and would therefore be an obstacle in Roderogo's plans to win her hand. Cassio had to be gotten rid of. He then told Roderigo to offend Cassio in some way and start a quarrel with him. He, Iago, would take care of the rest.

The scheming Iago then coerced Cassio into drinking some wine, even though Cassio had told him that he gets drunk much too easily and had already imbibed. Cassio is then in a drunken state and when he leaves for bed, Roderigo upsets him and they appear at the scene, fighting. The irony lies in the fact that Iago knew exactly why the brawl happened for the reasons mentioned above.

It also ironic that Othello repeatedly refers to Iago's honesty when he is anything but honest and is hell-bent on destroying both Cassio, Othello and Roderigo through his plotting, scheming and manipulation.

Dramatic irony is when a playwright creates a situation where the audience is aware of something that the characters in the play are not. Throughout the entire play, the audience is aware that Iago is manipulating the other characters without them knowing. An excellent example of dramatic irony takes place in Act Four, Scene 1. Iago attempts to convince Othello that Cassio has been sleeping with Desdemona by telling Othello to hide and watch as Cassio jokes about his affair with her. Iago tells Othello,

"Do but encave yourself, and mark the fleers, the gibes, and notable scorns that dwell in every region of his face. For I will make him tell the tale anew where, how, how oft, how long ago, and when he hath, and is again to cope your wife. I say, but mark his gesture" (Shakespeare, 4.1.71-77).

In reality, Iago plans to discuss Cassio's relations with Bianca because Cassio cannot help but laugh when he talks about her. Iago explains his plan to the audience by saying,

"As he [Cassio] shall smile, Othello shall go mad. And his unbookish jealousy must construe poor Cassio’s smiles, gestures, and light behavior quite in the wrong" (Shakespeare, 4.1.90-93).

While Iago speaks to Cassio about Bianca, Othello believes that Cassio is laughing about his affair with Desdemona. The audience is aware that Othello is being manipulated as he hides and watches Iago's conversation with Cassio. Cassio is also unaware that Othello is watching as Iago plots against him. After Cassio leaves the scene, Othello immediately asks Iago how he should go about murdering Cassio. Iago also convinces Othello that Desdemona gave her handkerchief to Cassio. Again, the audience is aware that Iago has acquired the handkerchief from Emilia and has placed it in Cassio's room.

Throughout the entire scene, Othello believes Iago's lies and accepts the story of Desdemona's infidelity under false pretenses. The dramatic irony creates suspense as the audience witnesses Iago's evil plans take root and influence the other characters to act upon his lies.

Irony of all types exists within Shakespeare’s tragic play. However, there are multiple examples of situational irony—irony that exists when something different happens than what is expected.

One of the clear examples of situational irony (because of its twofold irony) is in Act 5. Throughout Act 4, Othello has slowly been building to the climax of murdering his wife for her supposed unfaithfulness to him. Although Desdemona maintains her innocence to him and begs him to let her say at least one prayer before her death, in Act 5, Scene 2, line 95, Othello claims, right before smothering her, that “it is too late.” After this, Desdemona appears dead. However, she surprises the audience by speaking again thirty-five lines later by claiming her maid, Emilia, was “falsely murdered.” Yet the audience is in for another surprise when, after Emilia asks who killed her, Desdemona replies,

“Nobody. I myself. Farewell.
Commend me to my kind lord. Oh, farewell!” (138-139).

Desdemona’s absolution of Othello is surprising to the audience because she has just suffered an unjust death at his hands, yet she doesn’t wish anyone to know that and even calls him a “kind lord.” Such surprising words speak to her faithfulness to and love for Othello, even in the moment of her death.

In Act I, scene 1, I would identify a paradox in Iago's behavior as he puts his plot in motion. He tells Roderigo that he feels no loyalty to Othello, and that

In following him, I follow but myself.

But he then claims that "heaven is [his] judge." (I, 1, line 60). Iago seeks favor and promotion, but he wakes Brabantio and begins yelling accusations using vulgar language:

An old black ram is tupping your white ewe. (I, 1, line 90).

It's ironic in the extreme that a man not already reconciled to being a criminal or an outcast would act so recklessly and offensively as Iago does.

Much of Iago's language throughout the play harps on sexual matters. In Act II,scene 3, line 22, he ironically says to Cassio of Othello and Desdemona,

Well, happiness to their sheets!

In Act 3, scene 3, Desdemona tells Cassio that in her efforts to get him in Othello's good graces she will spare nothing, and

I'll watch him tame and talk him out of patience....
I'll intermingle everything he does with Cassio's suit.

This is ironic because it will be true, but not in the way Desdemona thinks. The pleas for Cassio will be overextended as she says, but the result will be to drive Othello crazy with jealousy.

In Act IV, scene 2, one of the most poignantly ironic speeches is made by Othello when, in his emotional agony, he says,

Had it pleased heaven
To try me with affliction, had they rained
All kinds of sores and shames on my bare head

If, he says, these terrible things had happened, he could have borne it. In effect, Othello is admitting that he has become his own victim, that the thing happening to him now—the presumed infidelity of Desdemona—is really not as bad as this multitude of other possible evils; and yet, he is allowing it to destroy him.

In my view, the most disturbing paradox or irony occurs in the final scene of the play. Just before he stabs himself to death, Othello recalls an incident in which he had, "in Aleppo once," killed a Turkish man who was an enemy of the Venetians. It is as if in his last breath Othello is still declaring his loyalty to the Europeans, in spite of all that has just happened and all they have done to him.

One prime example of verbal irony in Othello occurs when Iago and Cassio discuss the issue of reputation.  In Act 2, Scene 3, Cassio is upset that rumors have ruined his reputation.  To this, Iago responds:

As I am an honest man, I had thought you had receiv'd some bodily wound; there is more sense in that than in reputation.  Reputation is an idle and most false imposition; oft got without merit, and lost without deserving.  (II.iii.266-270)

Iago's comment is entirely ironic because the audience clearly knows that Iago's schemes are the cause of Cassio's troubles.  Iago is anything but honest as he has lied to several of the other characters in setting up his schemes.  Further, here Iago tells Cassio that reputation is not important; however, Iago has been bitter from the beginning of the play for what he feels is a slight to his reputation (not being promoted by Othello).  So Iago's comments to Cassio are a prime example of verbal irony.

Dramatic irony occurs when the audience has information that the characters in a play or work of literature do not.

In Othello, we as audience have been privy to Iago's innermost thoughts. Therefore, we know what the characters in the play do not: that Iago is an evil, malevolent, twisted person who wants to destroy Othello and Cassio. He thinks all people, women especially, are dirty and animalistic. Iago, however, is such a good actor and so good at manipulating people that nobody realizes what he is up to until after it is too late.

From that central dramatic irony all the other dramatic ironies in the play arise. For example, Emilia has no idea the dreadful damage she is doing to her friend Desdemona when she steals her handkerchief and gives it to Iago. Desdemona has no idea that when she innocently says kind words about Cassio in front of Othello that Iago has been whispering in Othello's ear that she is sleeping with Cassio.

The other characters are tricked by Iago because they are decent human beings who don't expect a person they consider a friend to hate them and want to destroy them.

The irony in Othello is that Othello believes his Desdemona has been unfaithful. In reality, Desdemona has been faithful to Othello. Iago has caused Othello to think otherwise.

Othello actually kills his beloved Desdemona for his thinking that she has been unfaithful

Iago has caused all the turmoil and anguish in the play. He is a jealous ensign. He longs for Cassio's position as lieutenant. Because of Iago's deceit, Othello becomes mad with jealously, thus suffocating his precious, innocent wife, Desdemona.

As is fitting, Othello takes his own life when he learns the truth about Iago. Of course, it is a little too late. Iago's damage has been done.