Othello's decision to choose Michael Cassio to serve as his lieutenant is the primary reason Iago seeks revenge. Iago believes that he is significantly more qualified than the young Florentine soldier and resents Othello for giving Cassio the esteemed title of lieutenant. Iago also wants to avenge Othello for sleeping with his wife, Emilia. Iago's racial slurs and derogatory remarks concerning Othello's race and ethnicity are additional motivating factors for him to seek revenge.

Iago also knows that Desdemona is the love of Othello's life and the person he adores the most. Iago desires to sleep with Desdemona in order to make things even, but Desdemona's loyalty to Othello makes this an impossibility. However, Iago reasons that it will be relatively easy to make Othello jealous and manipulate him into believing that Desdemona is sleeping with the younger, more attractive Michael Cassio. Since Iago cannot have Desdemona to himself, he is committed to destroying her happy marriage. Iago essentially uses Desdemona as a pawn in his scheme to ruin Othello. By convincing Othello that Desdemona is having an affair, Iago successfully ruins Othello's peace of mind and influences him to murder his beloved wife.

In Shakespeare's Othello, Iago is committed to destroying Desdemona. He realizes this is the most effective way to get back at Othello. Iago hates Othello. Othello overlooked him and named Cassio as his lieutenant. Iago is determined to get revenge.

He is disgruntled at having been passed over for promotion, and he sees a chance to get back at both Othello, who has slighted him, and Cassio, the mocking symbol of that slight.

Iago creates an evil plan to accuse Desdemona of having an affair with Cassio. In this way, Iago can drive a stake in Othello's heart. Othello is madly in love with Desdemona. Iago knows this fact. He uses Othello's jealous emotions toward Desdemona to get back at Othello. 

Iago will stop at nothing to see Othello suffer. He knows that the best way to get back at Othello is through the love of his life, Desdemona. Also, Iago hates Othello for his suspicion that Othello has slept with Iago's wife Emilia. For this reason, Iago is determined to use Desdemona to destroy Othello:

 I do suspect the lusty Moor [Othello]
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 evened with him, wife for wife.
(II.i.299-303)

Iago is determined to destroy Othello so he sets Desdemona up. Also, Iago manipulates other people in the play to get what he wants. Iago's ultimate goal is to get revenge for Othello's choice of lieutenant and to pay Othello back for sleeping with his wife.

Iaga has a plan to have Roderigo kill Cassio. Iago is evil. He wants Cassio's position. He uses Desdemona as a part of his evil plan.

Iago recognizes that Desdemona is kind. He encourages Cassio to speak with Desdemona after Cassio loses his position as lieutenant. As a master manipulator, Iago twists the facts to further his evil plot. He whispers suggestions to Othello to make it appear that Desdemona and Cassio are having an affair. Because Othello idolizes his wife, he becomes jealous to a  point of rage.

Iago knows that Roderigo also loves Desdemona. He uses this fact to turn Roderigo against Cassio. Iago wants Cassio dead. He sets up Roderigo in a plan to get Roderigo to kill Cassio. Desdemona is at the center of Iago's plan. He uses the love Othello and Roderigo have for her to advance his plot.

Desdemona dies as a result of Iago's evil scheme. Iago is one of Shakespeare's most evil villains. He destroyed Desdemona along with Othello. Desdemona means nothing to Iago. Ultimately, Iago is bent on destroying Desdemona because Desdemona is the closest person to Othello.       

In Othello, why does Iago want to destroy Othello, Desdemona, and Cassio—despite the fact that he already got Cassio fired?

This is, without hyperbole, one of the most written upon character-based questions in Western literature. There have already been a couple good answers on here about Iago’s anger that he was passed up for promotion.

His line early in the play: “I hate the moor,” also suggests a possible undercurrent of racist motivation. (A quick side note: Othello is meant as a dark Turkish character, but in Shakespeare’s time might not necessarily have been represented as “black” the way the part is today. Though it also very likely could have been played in blackface. Either way, the racist undertones persist.) Iago also states that he suspects Othello might have slept with his wife Emilia, though how deeply he believes this suspicion is open to debate.

Iago’s true motivation is a difficult question to answer. One of the fascinating things about his character is that he gives so many different reasons to hate Othello (listed above), yet scholars still feel sure there must be some deeper, grander root of his evil. The moment that separates Iago from the rest of the villains in Shakespeare’s canon, and indeed makes him the best of Shakespearian villains, is his own explanation of his misdeeds after he’s found out and captured:

Demand me nothing. What you know, you know.
From this time forth I never will speak word.

These are chilling last lines from a character who spent the majority of the play strategizing and using others as puppets. Scholarship might debate the question of Iago’s intentions for another four hundred years. But the mystery of his evil and his refusal to speak on them is more than a literary puzzle. The mystery of Iago’s intentions is one of the most engaging aspects of his character and is part of why he has endured as one of the best literary villains of all time.

In Othello, why does Iago want to destroy Othello, Desdemona, and Cassio—despite the fact that he already got Cassio fired?

Great question, and one which is never satisfactorily answered in the play. Why exactly is Iago so consumed by his desire to bring down not only Othello, but everyone around him, too?

At the beginning of the play, Iago says that he "hate[s] the Moor" because he has elected Cassio to be his lieutenant, even though Iago sees this as a gross mistake, as Cassio has never "set a squadron in the field," and is generally underqualified for the post. Iago feels that Othello has unfairly overlooked him, and as such, he wants to show Othello that he has done wrong to choose Cassio for this position. He succeeds in ruining Cassio's reputation and losing him his position, but the fact is, Cassio is not the only reason Iago dislikes Othello. Removing Cassio from his position only solves part of Iago's multifaceted problem.

Another reason Iago cites for his hatred of Othello is that he fears he "hath leap'd into [his] seat"—that is, slept with Iago's wife. But does Iago really believe this? We could argue that it's all part of a more complex trend of racism towards Othello: Iago talks about him using racist epithets, and sexualizes him to the point of describing his sexual conquest of Desdemona as being animalistic ("the old black ram is tupping your white ewe"). We could infer, then, that Iago's jealousy is intensified by the fact that a black man, Othello, is in a position of power over him to begin with, and moreoever is married to Desdemona, a beautiful white woman.

It isn't clear whether Iago really knew Othello would kill Desdemona—possibly he just wanted to drive Othello mad with jealousy and ruin their relationship. What is clear is that Iago's hatred of Othello takes on a life of its own: having set events in motion, he cannot step back from them, even when it kills Othello, Desdemona, and his own wife, Emilia.

In Othello, why does Iago want to destroy Othello, Desdemona, and Cassio—despite the fact that he already got Cassio fired?

Iago particularly wants to destroy Othello to seek revenge for Othello's naming Cassio as lieutenant.  Iago felt certain that Othello would give the title to him, so when Othello names Cassio instead, Iago is full of rage.  Further, Iago thus becomes jealous of Cassio because he is now of a higher rank and is young and handsome to boot.  Desdemona gets wrapped into Iago's evil plotting by mere circumstance--Iago knows that Othello values Desdemona and their relationship above all else, so Iago uses Desdemona as a tool in his plan. 

Iago becomes overtaken by his own jealousy and rage, so he continues with his plans even after Cassio is fired.  By this point in the play, Iago is no longer trying to get the rank--he is almost obsessed with seeing Othello's fall.