Here’s the thing: irony isn’t the absence of content, but the inversion of content. So we can still criticize the content of ironic statements as long as we consider their irony. Example: in Tommy Wiseau’s The Room, the protagonist constantly, and earnestly, greets other characters by saying “Oh HEY, [name]!” In the Rifftrax commentary to The Room, the Rifftrax guys constantly, and ironically, mimic the protagonist as greeting inanimate objects in the same way, e.g. “Oh HEY, sidewalk! Oh HEY, newspaper! Oh HEY, front door!” Whether you think this is funny or not, it has a definite meaning–to wit, that the protagonist’s method (and quantity) of greeting is bizarre.
That’s ironic. It’s “ironic,”* on the other hand, to e.g. tell a joke which turns on the implication that women’s inferiority and degradation, and then say that the joke isn’t sexist because you’re being ironic. That’s not to say that there are no non-/anti-sexist jokes about women’s inferiority etc.; Lindy West explains how to tell one here. My point is that there’s a difference between using actually irony to invert the meaning of a statement vs. just saying the statement more-or-less earnestly and then using the poise of “irony” to deny that you said what you actually said.
A shorter way of saying this: “irony” is not a free-pass to plausible deniability re: bigotry.
*That’s right: I’m using scare-quotes on irony. It’s ironic “irony.”