How about “June-Cleaver-Speaking-Jive” type movie?
On 2nd thought…
…I think I will have a second cup of coffee.
I’ve always called them “turkey movies”, with the early ZAZ stuff being seminal in the field. Amazon Women on the Moon would be an example of a non-ZAZ turkey movie.
FWIW, the term “turkey movie” was used by my cohort back in high school. I’ve encountered some people outside of that cohort that know exactly what I mean by “turkey movie” … and some that don’t.
“Zany”.
Not to be confused with “Laff Riot” which when spelled in that way, is to be avoided at all costs. 
They weren’t the first even in live-action film. The “A Fistful of Yen” segment in Kentucky Fried Movie was exactly the same kind of comedy, and there are probably other examples predating Airplane. On TV, there was the Mel Brooks sitcom When Things Were Rotten.
Especially the highly erotic shower scene.
I’ve used the expression “joke a minute” movie.
I nominate “Zazzer films”, in reference to the fairly large number of such films done by the ZAZ guys.
Ive said “rapid fire comedy”.
Call 'em anything you like – just don’t call 'em “Shirley.”
The OP seemed to be asking “what are these movies called,” not “what shall we call these movies?”
For twenty years, I’ve almost always heard them referred to as “Airplane!” movies. Everyone knows what that means; nobody seriously thinks you’re talking about Flightplan. Variations include “Airplane!-style” “Airplane! comedy,” when there could actually be some confusion. Occasionally, I’ve heard Naked Gun used in place of Airplane, but rarely.
“ZAZ” is occasionally used, but mostly by entertainment writers and people who wish they were entertainment writers. In any case, when “ZAZ” is bandied about, it seems to be understood that we’re talking about movies of the Airplane/Naked Gun genre.
The general terms just aren’t specific enough. Slapstick? So was Chevy Chase and Lucille Ball. Farce? So was Frasier. Rapid fire comedy? Dennis Leary, or Henny Youngman.
I proposed this: “Movies that make people trot out the same Shirley jokes we’ve been hearing since 1980.” Kinda wordy, though.
It would seem, then, that the only problem for using “ZAZ” is that it’s been limited to few people up to this point? Because it’s a really good word, catchy, rememberable and defined.
Slapstick, farce and surrealist humor all strike as “taken” as for what goes to comedy genres. Especially slapstick, that gives me Charlie Chaplin vibes.