Furry irony works in the same vein as hood or thug irony where furry content is used as a joke ironically. It is usually paired with unrelated pieces of text, traces memes with furry characters, or in a speech bubble format.
Person 1: Look at this furry irony meme!
Person 2: Dude you're a furry.
Person 1: Nuh uh. See? It's ironic.
Person 2: Ok man.