I've only ever heard of one Source for native Americans having a taboo against mouth breathing, and it was in an ask historians post. I have never seen a taboo against mouth breathing mentioned in any of the ethnographic literature I have read, or mentioned by any elder I've ever worked with, or mentioned by any counterparts in other nations. It may be that this was a thing or is a thing in some areas where the man from the original source traveled, but it doesn't seem to be the case in western Canada.

I have frequently heard, however, that Apache runners were expected to be able to hold water in their mouths during long runs. I've heard similar statements about other nations, usually in books related to endurance running. This would definitely have implications for mouth breathing, so hopefully somebody can chime in and give some alternate information relating to communities that did have such a practice.