A Google Books search finds only two instances of "it's not even funny" in its literal (or arguably literal) sense that antedate the first appearance of the idiomatic form of the phrase.
A headline in the Columbia Alumni News (October 20, 1922) proclaims "This Is No Joke!" with the subhead "It's Not Even Funny." And an advertisement in The Journeyman Barber, volumes 24–25 (1928–1929) [combined snippets] has this bit of doggerel:
D stands for DOLLARS
It's not even funny,
For, when you use Williams
You're sure to make money.
The next two matches, however, use the form of the expression that the OP asks about. From The Parchment (volumes 5–8) (1933–1937 [exact year uncertain]):
Lynn brushed the dark hair from his face with a restless movement, frowning. "I don't know. I don't even know what I want to do. I'm getting nowhere so fast it's not even funny."
And from a metrically atrocious poem called "Safety First" that appears in Trans-communicator, volume 53 (1936):
You may ride the planes or take the busses,
But you don't have room to settle little fusses.
You are packed and jammed till it's not even funny,
When room and comfort by train awaits for less money.
An Elephind newspaper database search finds this instance from "An Australian Girl Abroad: Radios Are Cheap in London: Midget Sets From Three Guineas Up" in the Narromine [New South Wales] News and Trangie Advocate (March 7, 1939):
Oddment Number Three. The indignation I felt over the programmes at the Earls Court Winter Cavalcade. In the first place they cost one shilling each—and when we arrived at the middle pages, we found one of them upside down—and the rest of the programme in that direction. That's a swindle, you know—I mean, it's not even funny. Two programmes, exactly the same, one right side up and one upside down, in a slick cover. And very little Information in it, anyway.
The implication of the expression, even in the 1930s, is the one that AHuman notes in a separate answer: Our first inclination when presented with a minor inconvenience may be to make light of it and perhaps to laugh at the absurdity of the situation or at the incompetence of those responsible. But as the level of unpleasantness increases, its seriousness becomes less and less escapable through laughter, and we finally reach the point where "it's so bad that it's not even funny."