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.
In "Othello" Iago is very jealous of Othello. He wants to be in a better position but he is out ranked by a Moor. He is angry that a Moor has power over him. Throughout the play he makes mention of Othello's color and is negative about it.
Then, Iago thinks that there is no way that a woman as lovely as Desdemona is going to love someone like Othello. Instead, she is captivated with him and loves him with all of her heart and soul. He again is jealous so he begins to set his plot to destroy Othello into motion.
Casio has also got the position that he had wanted and he knows a good way to cause havoc for all three.
Also, Iago believes that Othello has "twixt' my sheets. He's done my office" (Act 1 scene iii). He accuses Othello of having an affair with his wife Emilia and feel emasculated by this. "I do suspect the lusty Moor Hath leaped into my seat". Out of shame, he has decided that "nothing can or shall content my soul Till I am even with him, wife for wife" (Act 2 scene i)
That may be a reason why he continues with his evil plot even after Cassio has been dismissed.