Mr. Nolan once having Mr. Keating's job (Situational irony)
It's ironic that Mr. Nolan used to teach the same class that Mr. Keating is now teaching, since Nolan represents every tight-fisted, conservative value Welton holds, while Keating brings a fresh set of ideologies to his classes. By bringing to mind how different the class would be if taught by Nolan, the film underlines just how much influence Keating himself has on his students, beyond the content of his course, while also underlining the contrast between the two characters.
Mr. Keating Talking of a War (Situational Irony)
In one of his first lessons with the boys, Mr. Keating says, "This is a battle, a war, and the casualties could be your hearts and souls." This statement is ironic because it unintentionally foreshadows Neil's suicide, when Neil himself becomes a "casualty" in the war between his passions, inspired by Mr. Keating's teachings, and what his father thinks is best for him.
Keating Not Realizing His Own Impact (Dramatic Irony)
While Mr. Keating is aware of his intentional influences as a teacher, like encouraging the boys to be more daring, passionate, and independent, he isn't aware of their reformation of the actual Dead Poets Society, presumably, until after Neil's suicide. It's possible that he may have had additional advice for the boys regarding it, possibly even condemning it, the way he condemned Charlie's stunt with the phone call from God during the school assembly, and so his not knowing about it can be thought of as an example of dramatic irony.
The backfiring of Keating's influence (Situational Irony)
Mr. Keating's goal in his lessons was to inspire and uplift his students, encouraging them to be passionate and pursue their dreams. In the end, however, his influence has disastrous results, as the most direct effect of his teaching seems to be conflict between his students and the administration as well as the boys' parents, particularly in Neil's case. While Keating intended to encourage Neil to pursue his passions, his advice ultimately, if indirectly, led to Neil's suicide.
