Equality (dramatic irony)
Sgt. Hartman tells the new recruits to boot camp that he is a man all about equality. The irony is that he says this as he spits racial slurs and derogatory names at every race present in the barracks, displaying the exact opposite quality that he claims to possess.
Joker's Disavowal of the Virgin Mary (situational irony)
Joker tells Sgt. Hartman that he doesn't believe in the Virgin Mary, which angers Hartman, who asks Joker to take it back. Ironically, when Joker stands his ground, he is promoted to squad leader because he stands by his statement. Thus, Hartman responds in the exact opposite way we expect, rewarding Joker for disobeying his orders. Hartman respects Joker's ability to stand strong rather than backpedal and play it safe.
Punishment for the Donut (situational irony)
Sgt. Hartman finds a jelly donut in Pyle's lock box, against the rules which prohibit food outside the mess hall. Ironically, he doesn't punish Pyle, who committed the infraction; instead, he punishes the entire squad.
"Motivating" Pyle (dramatic and situational irony)
After getting constantly punished for Pyle's infractions, the rest of the squad decides to hold him down and beat him with bars of soap wrapped in towels in the middle of the night. Following Hartman's orders, they think that punishing Pyle will "motivate" him and thus bring him in line with Hartman's discipline. Ironically, though, the beating doesn't bring him into line, but breaks him down, eventually leading him to kill Hartman: precisely the opposite of following his orders.
