Due to audio requirements, its harder to do a site specific musical vs. a play. So they are usually rare.
Though not a musical in the traditional sense, The Donkey Show at ART makes great use of the club space at their Oberon venue. The actors are singing along to tracks of disco hits (which helps with sound issues/being able to hear everything). The direction is quite smart in the way it utilizes the setting and naturally moves the audience around the space.
They tried another musical in that venue, Prometheus Bound, which...wasn't great. Though Uzo Aduba had a killer song and Gavin Creel was there in his shirtless glory. Could barely understand many of the songs though, especially ensemble numbers. And some confusing directing/storytelling choices turned a straightforward story into something incomprehensible.
I saw an incredible version of The Threepenny Opera in Chciago, done by The Hypocrites. They used the garage space at Steppenwolf, which of course has been re-purposed for theatre. But it fits the site specific bill since its the bottom level of a parking garage and a very unconventional space. The score was performed with a sole piano in the center of the space, seating all around the edges and around two giant wooden tables (which performers frequently performed on top of. The stomps and claps in the choreography took the place of the other instruments in the orchestrations. It was pretty awesome.
Those are the only musicals I can think of off the top of my head. Any other site specific work has been plays. Which seem more easily transferred into unconventional spaces.