A statement is said to be tongue-in-cheek if it is not to be taken seriously. How did this meaning come into vogue? Where did it originate from?
Wikipedia suggests two meanings for the physical action. The relevant one:
This matches with what I have always assumed was the answer: You are biting your tongue, cheek or lip to keep a straight face. The intent isn't so much to cause pain but to distract you from the joke or bluff. Another common behavior is to purse your lips when trying not to smile.
An expression, and practice, which has largely fallen out of vogue (since the mid-to-late 20th century)—and for which the above Wikipedia definiton holds a limited measure of validity, "tongue in cheek" most accurately alludes to the act, or refers to the act itself, of surreptitiously pushing one's tongue, noticeably, into one's cheek on either side of the face, so as, while restraining any frontal lip or facial expression, to gesture (to signal) to one or more "confederates" in view of the "tongued" cheek, but conceal from another—usually the brunt of some joke or act of mirth—who is facing the opposite (untongued) cheek. It is tantamount to a wink (especially for those unable to wink), but further serves as a means, ala Wikipedia's biting of tongue, by which, when needed, to suppress any visible facial "tell" that might be "picked up" by the person intended to be deceived.
It is both the act, per se, as well as a spoken or written allusion to the act: of a thing that is less than sincere, for example, one might say it was "tongue-in-cheek."
Since no word etymologies are lacking, and the expression (and related act) falls within this writer's authoritative purview and experience, no further authority or source is needed or feasible.