Almost everything under 20hz is inaudible, so cutting it out of the track isn't too noticeable. If anything, it helps when compressing or limiting the track, if they don't have a HP roll-off.
As for the speaker damage rumour, it has some truth to it. If someone has produced a track without paying attention to the sub end (very common with new producers because you cannot hear it - out of sight/hearing, out of mind), and something's put a stupid amount of bass energy down there, like a loud kick-drum, it's going to smash the hell out of the subwoofer driver/amplifier, which CAN get damaged as a result. I vaguely remember reading that most big speakers/subwoofers have safety thresholds for damaging levels nowadays, however.