You poor lost soul. Your formative years were during the Disco Era and the TV series Dallas and the Love/STD boat? How do you manage to survive in these modern times? Is there some government program that helps people like yourself get along? 
Not quite what the OP was asking, though maybe the brain/memory is playing some tricks here, but wasn’t Johnny Carson of the Tonight Show famous for saying that as well?
I think the roots of this go far back into vaudeville for the comic usage. I’m going to pose this in GQ to get the etymologists to work.
It doesn’t sound so much like Groucho as Chico. In fact I can see the scene, with Chico on an old candlestick phone.
You don’t say.
You don’t say!!
You don’t say!!!
(Groucho) Who was that?
He didn’t say.
I hasten to point out that just because I can imagine the scene doesn’t mean it ever actually took place.
Rosemary the receptionist in “Hong Kong Phooey”.
Yes, I am a cultural polymath.
First time I ever heard the joke was an episode of Hong Kong Phooey.
You were probably thinking of Happy Go Nutty, a Screwy Squirrel short. The “you don’t say” bit is at about 3:49.
I could be wrong but I doubt you’re ever going to find an original source, and I think focusing on the vaudeville joke is putting the cart before the horse – the twist comes with the final line, based on the set up, which implies that the usage in the setup was already a current speaking convention.
That is, by the '30s at least, “you don’t say!” was a common response to some surprising or unusual piece of news.
That seems to be a common formation in a lot of languages – inviting confirmation by saying something along the lines of “is that really the case, because I might have expected otherwise?” We do that too in a slightly different way when we say “Really??”, indicating our surprise by implying that the statement previously made was a bit surprising or implausible and needs verification.
I suspect many languages, maybe most, have developed similar forms. In Japanese they’d use “hontou?” (really?) or maybe a couple of others I’m blanking on. I doubt anyone’s ever succeeded in attributing such usages to a single source in those languages either.
That’s the right era and style, but I have no recollection of “Screwy Squirrel.” Although it’s possible I’ve seen that short in one of those old-style photo-booth looking things they used to have over at the K-Mart in the 80s, where you can put in a quarter (or was it two?) to watch a 3-minute cartoon.
I always picture Edward Everett Horton wide eyed and rubbing his chin.
( Arsenic and Old Lace ? )
May I put forward Jerry Calonna (I think that is the spelling). The pencil thin moustached and palmaded Calonna’s most common tag line was “Yesssssss (with rising inflection),” after appearing in a service window (or something similar) of some sort. However, he also regularly said “You don’t say? (with almost a squeak at the end)” while rolling his eyes.
He was a regular second banana to such comics as Benny, Hope, Abbot and Costello and the like.
OK, it’s Colonna and here is the wikipedia cite:
I can recall Bugs doing it in that nasally drawl; the earliest I remember is one of the two crows doing it in a 16mm Heckle & Jeckyl cartoon that my brother found (with the projector) in the attic. It was pretty ancient back then…
I think I remember one of the stooges doing that, as well.
This routine shows up during a Spike Jones performance of the song “Chloe” on the radio program Bob Burns Show aka Lifebuoy Show of October 28, 1943.
Already noted upthread. Which was about 3 years ago…
I came to suggest Frank Nelson and agree this sounds conflated.
I’m pretty sure someone did it in an old Looney Tune at least once (maybe Daffy Duck, probably just as Elmer Fudd was about to pull the trigger on his shotgun).
It’s also the punch line to an old joke about a wide mouthed frog, as recaptured in this commercial.