My old rule was that a musical has a busy enough soundscape with music and singing and dialog (and footsteps and audience sounds and so forth).
But I just did a second show in which I had background effects running through many of the songs. And this was a kids show, too (that is, a young cast, not, a show aimed for kids.)
In "The Little Princess" I ended up with a wind loop that ran under at least one song. Also a rain loop and some added thunder for another.
For "Tarzan" I actually had jungle noises (mostly birdsong and insects) playing through many if not most of the numbers.
Now, this was a young cast. But for some reason I had a ton of headroom in the mix, and clarity to spare. An ultra-small pit of keyboard and drums (the drummer also played Djembe, and the keyboard player was often in a split patch with bass under his left hand). They played soft and controlled, and I've been tinkering with the system, and I had no trouble getting clear mic signal above it.
And, well, it is mostly Phil Collins. Which means open ballad-like songs with a lot of space in them. Space for more orchestration. Or space for effects.
The key, as always, is ears in the house (aka a front-of-house mixing position), and a fader so the background effects can be trimmed moment by moment just like backing vocals or the drum mics.
Hollywood does this kind of dense design these days. They can, of course, because they are cutting to the final print. In theater, you are up against the spontaneous nature of the thing. You can't so closely predict where the spaces are going to fall in dialog, and where you need to hold back on the underscore.
Also, Hollywood is made of close-mic. The ability to isolate, compress, and stack up dry sounds very close to each other. Theater is an acoustic environment. The voices of the singers, the backline monitor amps of the band, the sounds made by the audience all go into that live reverberant space. Even when you have microphones, you are often constrained to place them further away -- meaning again a more distant, muffled, less precise sound.
But when you come up against a musical like "Tarzan" (or to an only slightly lesser extent, "A Little Princess") you are dealing with a sonically artificial setting. Pop songs, studio sound, not the invisible reinforcement expected of a classic Rogers and Hammerstein. And that kind of sound -- with the strong amplification implied -- gives you room to also treat sound effects in a more immersive way; more movie, more gamescape, than would be traditional for theater.