If only someone had thought of it two posts before yours!
Roger Ebert labeled Fargo as a dark comedy, so I don’t know if you’d consider it “funny” or not (but I still laugh at parts of it).
If you also like Classic Movies: To Catch a Thief.
It stars Grace Kelly and Cary Grant; directed by Alfred Hitchcock.
In a way it’s an anti heist movie.
From IMDB:
When a reformed jewel thief is suspected of returning to his former occupation, he must ferret out the real thief in order to prove his innocence.
The one that jumped first to my mind was Topkapi with Peter Ustinov.
Sam Whiskey with Bert Reynolds, Ossie Davis, Clint Walker, among others comes to mind. An old west comedy where a group tries to break into the Denver Mint.
Most of my favorites have been mentioned, but there’s also
How to Steal a Million (Audrey Hepburn and Peter O’Toole!)
Topkapi (Melina Mercouri, Peter Ustinov, and Maximilian Schell, among others)
Thief – Michael Mann movie with James Caan (and Willie Nelson!) – not funny, but interesting. And Mann seemed to mine it later for his TV show
All the Oceans Eleven etc. movies.
City Heat is a caper comedy from the police’s perspective.
Along similar lines: Duck, You Sucker (aka A Fistful of Dynamite). Not a straight-up comedy but it has its moments.
I know, right? 
I saw that the other day on TCM. And the same studio also made The Lavender Hill Mob, which was also great.
Wow–I’m taking notes and writing down a ton of these.
I’ve seen Mad, Mad, World and What’s Up Doc?* and I don’t know if I’d think of them as caper movies, but they’re among my favorites and are great for “calibrating” what others are thinking of as funny.
I haven’t seen most of the others listed (I saw the modern Ocean’s Eleven and the Alex Guinness “Ladykillers” and liked 'em both), so I really appreciate the recommendations: please keep 'em coming! 
*
Ryan O’Neill: It gets kind of complicated now. First, there was this trouble between me and Hugh.
Judge: You and me?
Howard: No, not you. Hugh.
Hugh: Aye yam Hugh.
Judge: You are me?
Hugh: Noh! Aye yam H-yoo.
Judge: Stop saying that!
Judge (to baliff): Make him stop saying that!
(Baliff advances on Hugh)
Hugh: Don’t touch me, I’m a doctor.
Judge: Of what?
Hugh: Music.
Judge: Can you fix a hi-fi?
Hugh: Noh
Judge: Then shut up!
:mad: There is nothing funny about a Mini falling off a cliff, I assure you!
Here is your winner:
If you enjoyed that, track down Kind Hearts and Coronets. Kind Hearts and Coronets (1949) - IMDb
The Freshman. The whole movie is basically about one extravagant thread in a swindle/heist. One of my favorite movies.
A wonderful Italian film that predates virtually everything already mentioned: Big Deal on Madonna Street.
The Thief Who Came to Dinner starring Ryan O’Neal had its moments. Decades after it came out, I still remember one or two memorable scenes. It is very dated today, but Ryan O’Neal had a warmth that was charming and the schemes were fun.
Sneakers.