"Even the sardines taste different..." (dramatic irony)
In the sequence towards the end of the film when Cop 663 has developed a new lease on life and decided to get over his stewardess ex, there's a shot of him eating sardines in his apartment. As part of his newfound appreciation for simple pleasures, we hear him say in voiceover, "even the sardines taste different now." Little does he know that the new taste is actually due to the fact that his sardines are likely expired, as earlier we had seen Faye switch around the labels (and therefore expiration dates) on all of the cans of sardines in his pantry.
Cop 223 falls in love with a drug smuggler (situational irony)
On one hand, it's simply ironic that Cop 223 falls in love with the woman in the blonde wig because he's a cop and she's a drug smuggler. The irony illustrates that he is an especially incompetent law enforcer, who is so focused on finding love that he misses red flags. This situation suggests to us that he misses these red flags both on the job and in his personal life. But it's also ironic that he falls in love with her after colliding with her in the midst of a foot chase. We therefore watch him pursuing two criminals, although in two very different senses.
The California (situational and dramatic irony)
The end of the movie is a little confusing with all of its plays on the word "California," but this confusion is likely the intentional result of some ironic juxtapositions. Officer 663 invites Faye on a date to The California, but she has instead left town to travel to the geographic location California. It seems like a funny miscommunication, until later we learn that Faye went to The California, but potentially a different bar named The California. The missed opportunities that spring from this are themselves ironic insofar as Faye previously staged so many so-called chance encounters in order to interact with Cop 663 outside of Midnight Express.
April Fools Day (dramatic irony)
April 1st is the day that Cop 223's girlfriend broke up with him, and he has half-convinced himself that it came as some sort of prank. The irony here is that Cop 223 thinks (or really, hopes) that his girlfriend is playing some trick on him, but the person who's fooling him is really just himself. He waits until May 1 for her to come back, sulking all the while.