You wouldn't want to do the headphones thing because they require wires and that would get in the way of eating, you'd have to have a place to store them at the tables, and then it totally knocks out the family experience thing because you single yourself out to hear only the movie (no communication).
Outside, Muppet topiaries line the wall, there's a huge sign that has the restaurant logo on it, there is a crashed SwineTrek on the roof with lights and smoke, there are murals of Muppets and posters of Muppet movies around the walls. The exterior looks a little like a movie theater, or at best The Muppet Theater.
I see a Muppet-themed menu like has been mentioned, the tables are clear tops with pictures of Muppets and Muppeteers from over the years (including Sam and Friends, Wilkins/Wontkins, etc), The chairs are all different styles. There are Muppets on display behind glass, there are larger props and other memorabilia around the top of the room and hanging from the roof which is painted like a sky, and an animatronic Gonzo holds baloons and floats over the guests every hour (he goes right over the tables down the center and goes over the props around the ceiling). There are TV's around and speakers everywhere, and when a song is being pumped through the sound systems there is psychedelic things on the screen (like the end of MFS), but then a 'music video' plays on the screens, which are songs (at least sketches) from various Muppet productions.
There's a window by the regular kitchen's door that you can see an animatronic Chef preparing food, with the occasional chicken popping out of a pot on the stove, talking veggies in the fridge which opens suddenly by itself, talking cakes in the ovens, penguins walk through. It's a large window that can be seen from every seat in the house, and has a blind over it upon which a spotlight shines about the time it's going to play its little show. There are 4 different shows and happens an hour apart from one another. This would last maybe 5 minutes if that much.
There is a stage that looks like the Muppet Theater with the Box Seats...like at MuppetVision 3-D, just a smaller scale. It is also animatronic and there is a Cabaret styled Muppet Show that takes place, the EM Band in the background of the stage, the action of Kermit, Piggy, Fozzie, Gonzo, Rizzo, Camilla, Scooter, Pepe, Clifford, Penguins, etc happens in the foreground. This would happen every 2 hours, and it is timed to perform 4 or 5 different shows which last 15-20 minutes.
The sound system pumps through the Chef and Muppet Show instead of the regular line up of songs/videos. While Gonzo flies overhead you hear him say things like, "It's so COOL up here! Hey, down there! Chew your food! Clean your plate! That looks good! How are you?"
That's what I think.