I never investigated any official etymology of the term, but I’m pretty sure it comes from denigrating those who are unable to conform to a more socially-acceptable nose-breathing method, or are unable to care about the social standard that breathing through one’s mouth apparently violates. My younger sister has Down syndrome, and for reasons too complicated to explain here, there was a period of time during seventh grade when I rode her bus (the “shortbus” of grade-school legend) to school. My sister and I would get on, the bus would pick up several more mentally-challenged students, drop them off with my sister at their special-education elementary school, then drop off me and the junior-high-age “mainstreamed” special-ed students at my school. Oddly (in retrospect), nobody at school ever teased me for riding the shortbus to school. Anyway, my sister has always had trouble breathing through her nose, and most of the kids on the bus were mouthbreathers as well, either because of biological necessity or because telling them to close their mouths for social propriety’s sake was a hopeless exercise in futility. Didn’t matter, of course, since they got the oxygen they needed, and there were plenty of other social proprieties that took far higher priority.
Still, the times when the term “mouthbreather” was used around me, it was generally an insult used to imply that the target was the sort of kid who rode the shortbus, couldn’t tie his shoes, usually didn’t notice the drool on his chin or the booger in his nostril, sometimes made loud and incomprehensible sounds, and was oblivious to the social niceties of polite company. It’s basically the same kind of insult as when formerly medically-specific terms as “moron” and “idiot” were used to describe people who were, to employ an oversimplification, stupid without a good reason, as opposed to people with mental handicaps beyond their ability to overcome, which would hopefully have inspired more compassion.
So it does have an ableist background, I feel certain, and its use should be avoided as completely as using “retarded” as an insult, if one does not wish to inadvertently give offense.
I have uncommonly capacious sinuses, and boy you wouldn’t believe the volume of nastiness I can blow outta my nose on my snottier days. As a kid, sometimes it was easier for me to breathe through my mouth, so it became a frequent unconscious habit which my mother often tried to get me to break. To this day I try to remember to breathe through the nose when I’m able, and I’m always terribly self-conscious about possible halitosis, and I honestly have no idea whether nose-breathing or mouth-breathing is healthier (if all things be equal and the boogers don’t obstruct), but I know my own self-consciousness about it, when it occurs to me, is based on an underlying fear of appearing to be someone who is either ignorant of the public’s prejudice against mouth-breathing, or simply doesn’t care enough to meet the social standard.