Ironic Facial Hair is novelty facial hair grown with the intention of being "ironic"... Although the results are usually about as ironic as that song by Alanis Morisette. It is generally considered a hipster term.
Ironic Facial Hair can either be subjective toward the bearer (for instance a Jew with a toothbrush moustache) or more commonly achieved by growing a non-conformist style of facial hair that is rarely seen in modern society (for instance, the salvador dali moustache or mutton chops) the latter making this applicable to the hipster community.
"I had a Rap Industry Standard goatee BEFORE they were cool"
"I'm so non-conformist I'm going to grow some Ironic Facial Hair. I can't decide between 'The Super Mario' or 'The Jack Sparrow'"
"Dude, they're both way too mainstream - get a 'Franz Josef'"
Mel Gibson's sinister-looking imperial "evil villain" beard at the 59th Annual ACE Eddie Awards in 2009 was incredibly ironic.