Well, for the most prominent example, see what happened when Europe abandoned common sense in favor of comforting leftist platitudes and threw open the floodgates to uncontrolled mass immigration, with any sense of self-preservation dismissed as “racism.” Crime rates have skyrocketed, especially (though not only) rape. Offhand I’d say that that seems damaging.
Political Correctness itself is dangerous in the sense that rust is dangerous. Rust isn’t dangerous or scary. You can hold a small pile of iron oxide in the palm of your hand and chuckle to yourself at the idea of anyone finding it threatening. But watch what it can do to a car if left untreated. That’s the relationship between PC culture and Western civilization.