Tony taking Loscudo's Hat (Dramatic Irony)
In the beginning of the film, we see Tony working at the Copacabana. A very wealthy patron, Mr. Loscudo, gives his hat to the coat check girl and tells her to guard it with her life, as it's very important. Hearing this, Tony decides to steal the hat in hopes that he will be able to return it to Mr. Loscudo later and make himself look like a hero, which will bring him a handsome tip. When Loscudo finds the hat missing, he is absolutely furious, but Tony comes to his rescue by returning it, and the plan works, as Loscudo gives him a big tip. This is an instance of dramatic irony, in that the viewer knows Tony is the one who stole it, but Loscudo has no idea.
"Of course not" (Dramatic Irony)
When Don is getting beat up at a bar, Tony saves him by telling his assailants that he has a gun in his pocket. The bartender gets upset and orders Don's assailants to let Don go, lest Tony shoot one of them. Outside in the parking lot, Don asks Tony if he actually has a gun and Tony says, "Of course not!" In this moment, the viewer knows something that Don's assailants do not.
Tony had a gun after all (Situational Irony)
The gun irony becomes all the more ironic when it turns out that Tony actually does in fact have a gun. After their dinner in Birmingham, Tony fires two shots into the sky as they walk towards the car, to scare away some men who are hiding behind the car planning to mug Don. In this moment, we see that Tony had a gun after all, but just lied to Don about it earlier.
The cop wants to tell them they have a flat tire (Situational Irony)
On their final drive back to New York City through a snowstorm, Tony and Don get pulled over by a cop. Given their previous bad experiences with racist cops, Tony and Don are prepared for the worst, but the scene takes an ironic twist when the cop simply wants to let them know that their car has a flat tire.