I’m thinking Hitchcock. Maybe Rear Window, keeping James Stewart. There aren’t a lot of humans to replace with muppets, but I think it’d work.
Or maybe more absurdist. Raising Arizona, keep the human baby. Sweetums could be the hunter… Smalls, was it?
For the Coens, I was thinking Fargo. Kermit as Jerry Lundegard, Piggy as his wife, Gonzo in the Steve Buscemi role. Keep Frances McDormand as Marge.
And really, you could just drop Gonzo into any Steve Buscemi movie.
They did a very brief “tribute” in a 1981 special. https://www.youtube.com/watch?v=-BCswxH6VNI
Nah, man…Arnie has to be replaced by a Muppet - Link Hogthrob would be a good choice.
Make Kyle Reese the only human.
Miss Piggie as Sarah Connor.
And keep the Kyle/Sarah love scene.
(Terminator’s significant cast is too small to really be fun for this, though…)
Hmm…
Not originally a movie, but it’s been filmed, so it fits.
Jesus Christ Superstar:
Significant (having at least one feature scene as a soloist)
Jesus - Kermit (natch).
Mary Magdeline - Piggie (also natch).
Judas - Hmm…I think Gonzo would work best.
Peter - Fozzie.
Simon Zealotes - [Edit - spent too much time thinking about this, moved on… Then forgot to backtrack and come up with one. Oops] The human choice, since I can’t come up with a Muppet to fit, unlike all the others. Time travel to grab Larry Marshall.
Ciaphas and Annas - Tempted to go with Bunsen and Beaker, but… No. Making Annas’s lines unintelligable would require too much writing around them. Let’s say Pepe the Prawn and Rizzo the Rat.
Pilate - Sam the Eagle.
Herod - Rolf (Herod’s song is pretty piano-heavy).
Minor, but significant roles -
Money Changers - The Electric Mayhem
Other priests: Bunsen and Beaker can be here.
Not-in-the-original-movie:
Statler and Waldorf giving commentary during the 39 lashes.
Does the lone human have to be the same actor that played the role originally, or can we replace them with someone better at working with Muppets?
Jaws. All the formerly human characters are muppets, but it’s a real live shark.
The Blues Brothers with Kermit as Elwood, Fozzy as Jake, Dr Teeth in the Ray Charles Role, Miss Piggy in Karrie Fisher’s role, Janice could take Aretha’s spot, the rest of Electric Mayhem can fill in as the rest of the band. Get Steve Carell to play the leader of the Illinois Nazis.
Replace the entire cast of E.T. with muppets. E.T. himself to be played by James Earl Jones. If Jones isn’t available, then Clint Eastwood as he was in the 70s.
OK this is totally cheating and doing it wrong, but I would love one episode of a Gordon Ramsay show where he is replaced by The Swedish Chef but everyone else stays as they are. 
Julie Andrews stays, in The Sound of Muppets.
Charlie Sheen stays, and Gonzo takes the title role, in Being John Malkovich.
Yes, you’re probably right, it was the first thing I thought of while writing the OP 
I’m pretty sure that’s fine (I can’t claim credit for this idea, I read it elsewhere) 
The Muppet Centipede, with Dieter Laser (the Mad Doctor) still keeping his role, although you can substitute a more famous crazy guy in that role if you want.
The Swedish Chef should be in the front, his speech as incomprehensible as ever*, Miss Piggy in the middle, and Kermit at the end.
I look forward to the scene where Dr. Heiter explains the joining process using his PowerPoint presentation, except now you can see a human arm snaking up through all three conjoined Muppets.
*You can substitute Beaker, if you want.
Princess Bride - keep either Prince Humperdink or Count Rugen as human. Rizzo as the other one? Or Animal.
Kermit as Westley, Piggy as Buttercup, Sweetums as Fezzick is a lock, Great Gonzo as Montoya. Pepe as Vizzini or Fozzy? Swedish Chef as the Priest, Bunsen as Miracle Max, maybe, but who for his wife? Or maybe Statler and Waldorf for that pair? I like Beaker for the Albino.
Harvey Keitel is the only human left in Thelma & Louise. The title duo are played by Janice (from the band) and Miss Piggy.
The biggest problem with this premise is that you almost have to pick a movie with an overwhelmingly male cast, because there are so few female Muppets. Miss Piggy obviously has to be the female lead, then you have Janice, then… who else?
O
M
G
Does The Silence of the Lambs keep Anthony Hopkins, or Jodie Foster? I’m not qualified to decide.
Don’t do Austin powers – go straight to the original. Do Goldfinger. Keep Sean Connery as Bond – either use his footage from the original, or CGI him in like Peter Cushing and Carrie Fisher in Rogue One.
Dr. Benson Honeydew as Auric Goldfinger
Both Waldorf and Statler as “M” (the same way they were The Marly Brothers in Muppet Christmas Carol)
Miss Piggy as Piggy Galore
Janice as both the Mastersons
Bobo the Bear as Oddjob
Fozzie as Felix Leiter
Gonzo as Q
Camilla Chicken as Miss Moneypenny
Rizzo the Rat as Solo
I realize this leaves us with no place for Kermit. If it were all-Muppet, of course, he’d be Bond, but if we have to have a human, I’d prefer Sean Connery be it, rather than Gert Frobe or some other stand-in.
Off the top of my head, there’s Denise, the other pig Kermit dated from the recent series (AKA Natalie Dormer-as-a-pig); Jill the frog from Manhattan; Camilla, Gonzo’s chicken paramour; Miss Poogy; Yolanda the Rat.