There was no Ransom (Situational Irony)
The Dude and Walter lose the ransom money after the Dude's car gets stolen, and they spend a great deal of time trying to track it down, getting into outrageous situations along the way. Ironically, at the end of the movie, they realize that there was never any money in the briefcase the entire time, that Lebowski had made his own "ringer" to give to the ransomers, using this as a way to cover up his own embezzlement.
The Toe (Dramatic Irony)
Before the Dude figures out exactly what's going on, he stands on the side of the road, and the camera follows a passing red convertible, driven by Bunny Lebowski. The camera drifts down to her feet, showing that she has all her toes, which clues the viewer in to the fact that she is not the person whose toe was sent to Lebowski; she was not kidnapped after all. In this moment, the Dude does not realize this, so the audience knows something he does not.
The Dude's Assailants (Dramatic Irony)
Early on in the film, the Dude arrives home and walks through his front door. For a split second, the audience sees the two men who have broken in to the apartment come up behind him and attack him. In this moment, there is dramatic irony between Dude's cluelessness and the viewer's perception.
Larry's Corvette (Situational Irony)
One of the more humorous instances of irony is when Walter destroys the Corvette outside Larry's house, which he believes to belong to the young boy, and which he believes Larry purchased with the stolen million. He takes a crowbar to the windshield and destroys the car, yelling, "This is what happens when you fuck a stranger in the ass!" Ironically enough, the car that Walter is destroying does not even belong to Larry, but to an agitated neighbor.