What I'm thinking of is the recent case when the Italian Prime Minister in the European Parliament got angry at a German MEPO and said he'd make a fine concentration camp guard. When the place erupted he tried to wriggle out of it by saying that he'd been talking ironically, implying that Germans didn't understand irony (ie hadn't got a sense of humour.) But very clearly he wasn't being in the least ironic.

It's very common for people making racist or sexist cracks to claim "it was just irony" when they get challenged, and to indicate that anybody objecting clearly hasn't got the naus to appreciate subtleties like that.

The essential thing in irony is that the surface meaning is actually the reverse of what you appear to be saying. It's not the same as exaggerating for effect, or saying something that you don't mean absolutely literally.