So what the hell am I remembering - a scene where someone is getting a shave, and the barber hands the razor to someone else?
In Mississippi Burning Gene Hackman and Brad Dorif deliver one of the better scenes ever. Both contribute, in much the same way that Dennis Hopper and Christopher Walken are both needed for that True Romance scene (which is not a barbershop scene – but good anyway,)
Stickmen (A New Zealander movie about a underground pool tournament) has a super funny scene where the main bad guy (or what passes for a main bad guy in this movie) is making a business proposition to one of the main good guys while giving him a haircut and a shave… using a straight razor… and the guy giving the haircut had no hands, just two sets of those prosthetic hooks. All the while the barber guy is complaining about how he’s no good at threatening people. 
I’ll give you an obscure one.
The Apple Dumpling Gang.
The town sheriff played by Harry Morgan was also the town barber and the town Justice of the Peace. He was in the middle of shaving a customer when a couple came in to be married. He was so happy he kicked the guy out in the middle of the shave and only charged him half price.
No, I havn’t seen this recently, just got a great memory for stupid things.
I seem to remember one in Unforgiven
In The Godfather, the hitman, the one who testifies in The Godfather II, is getting a shave in a barber chair immediately before he goes and whacks Moe Green, Barzini and a IIRC another one.
Is that the “This side needs a little snoop. Now just a little snoop on this side to even it up.” etc.
Walt Disney’s “Peter Pan” has a scene with Smee giving Captain Hook a shave. Funny stuff!
High Noon has a good barbershop scene.
Rounders has Ed Norton and Matt Damon getting shaved after a rough night at the tables.
The Music Man prominently features a barbershop quartet, which isn’t the same thing, but what the hell.
Oh, and The Women features an extended beauty parlor scene.
High Plains Drifter
(sings) “All you need’s a little faith, and see-through push-up bra!”
I think we give-a him a snoop too much, yes.