You know, with some of the weird film and TV show properties to take off as musicals I honestly can't say whether or not this would fail. My instincts tell me it almost certainly would, but anything seems to be possible in Broadway these days.
You got a point there. Who would've figured you could get a hit musical out . . . .
1) an old Roger Corman movie about a man-eating plant?
2) the charismatic wife of Juan Peron of Argentina?
3) a serial-killing barber who bakes his victims into meat pies?
4) a bunch of old poems about cats?
5) a former nun escaping from Nazis?
Who knows? Maybe AMOK TIME: THE MUSICAL could work.
Of course, the thing about all of those stories you just cited is that they're complete stories in themselves, with beginnings, middles, and ends.
Star Trek is a setting, not a story, and therefore lacks a beginning, middle, and end. A
Star Trek musical would by necessity be unable to encompass the entire
Trek saga -- it would have to tell a single story within the larger
Trek universe. But the problem with this is that the power of a given
Trek story is derived in part from its relationship with other
Trek stories (be they episodes or films); this is true even of the new film. A stage musical's power, on the other hand, is supposed to come from its own terms -- it is supposed to be a self-contained unit. The sources of creative power are incompatible.
Further, much of
Trek's appeal comes from its fantastic setting -- one that the live stage would be hard-pressed to successfully re-create, in either a Naturalistic or non-Naturalistic design aesthetic. There's also the camp factor -- they're on a spaceship,
and they're singing. Musicals work best when they either insist that you take the fundamental conceit of the genre -- that everyone will sing their feelings -- absolutely earnestly, or when they lampshade the absurdity of the concept. It would be hard to convince the audience to buy
one fantastic conceit -- that of singing -- when you're already being asked to buy
another fantastic conceit -- that you're in space. Combining the two would be hard to do earnestly, and would be inappropriate for
Trek to do as a gag.
Most of all, though, I think
Trek would be hard-pressed to come up with the kind of story with the emotional punch necessary to sustain a large-scale musical. That's not to say that
Trek can't be emotional. But
Trek's power comes from intellect as much as emotionalism; its emotions are usually tempered. In musicals, emotion dominates. I suppose you could do something like, say, "The City on the Edge of Forever" as a musical -- but, then, if you do, you're basically doing a musical set in 1930s New York, and the
Enterprise almost seems superfluous to the musical's purposes.
I studied opera, so I have to say that. 
I do love musical theater though, I truly do, it's what got me into the theater. However, Star Trek as a musical?! NO!!!!
