See you in the funny paper[s] means "Goodbye, see you soon".
A Dictionary of Catch Phrases (1986) by Eric Partridge and Paul Beale says:
see you in the funny papers (—often and orig. I'll). 'This jocular farewell suggests that the person addressed is rather laughable: US: 1920s; extinct by the 1950s' (R.C., 1978). Perhaps adopted in the UK from American servicemen c. 1943. By c. 1955, (I'll) see you in the funnies.
The OED has funny paper from 1874 and funny column from 1860, meaning "a (section of a) newspaper containing humorous matter or illustrations".
Here's a 1922 example I found of the phrase in the signoff of a report in The Tusla Scout from Troop 12 by Ed M'Lain (published in The Tulsa Daily World, March 19, 1922):
Good-bye, see you in the funny paper.
This use by scouts suggests it's not insulting, but may be used in a good-natured, light-hearted mocking manner.
I found some 1920 and 1921 snippets in Google Books. The years need verifying, but here they are for completeness.
A University of Virginia student paper The Virginia Reel, Volume 1, 1920:
Well, boys, must close now. La, la, till the next time, and I'll see you in the funny papers. Ever your, ADELAIDE.
Commercial Telegraphers' Journal, Volumes 18, Commercial Telegraphers' Union of America - 1920:
So long, boys, see you in the funny paper. "30."
J. N. HANNA,
Union Postal Clerk & The Postal Transport Journal - Volume 17 - National Federation of Post Office Clerks, United Federation of Postal Clerks - 1921:
We will see you in the "funny paper" next month.