The three earliest occurrences in Google Books search results are from the late 1950s, and they seem quite random.
From American Flint, vol. 47 (1957):
Changes are being made every day around the plant—it sure seems good to see a clean and clear floor space. Hey Gerald, sweep over in that other corner—you missed a spot.
Well, enough stuff this time. Next time more. Remember to buy American, think American, and don't ever forget that you are an American, and darn proud to be so!
From Finlay McDermid, See No Evil (1959) [two occurrences]:
It scared the hell out of me when a voice called down from the oak tree, "You missed a spot, boy." I couldn't have been more startled if I'd seen the real Cheshire Cat when I jerked my head up, instead of a grinning little girl...
"You missed a spot, Benjie. Run your mower over toward me."
From The Gopher: Annual Publication of the Student Body of the University of Minnesota, vol. 72 (1959):
You missed a spot, pledges. See that you stay alert and complete your projects with greater precision in the future. Service projects receive more than the usual collegian's attention from the members of Phi Epsilon Pi fraternity.
As these examples suggest, the phrase "you missed a spot" tends to come up in the context of a nit-picking supervisor looking for a shortcoming in a subordinate's work as a basis for teasing or criticizing the subordinate. (For some reason, military service seems like a natural environment for the phrase to flourish in.) The supervisor may be actual, in which case the fault-finding is likely to result in demerits of some kind levied against the subordinate, or pretend, in which case the phrase is usually jocular.
Having said all that, I don't think that using the phrase is funny, either.