In the “It will never, ever happen, but damn, wouldn’t it be nice if it did?” category, I’d love to see Fiorello!. It won a Tony and a Pulitzer and other than one very nice cast recording, nothing survives of it: no other recordings that I know of, no films, etc.
While we’re at it, Greenwillow would be great but would never be done.
RENT retold from the standpoint of Bennie and those who actually pay home and shop rent in Alphabet City where the main characters are squatters. I’d call it EVICTION.
I wonder if there is a workable english version of “Falco Meets Amadeus.”
Oh! Calcutta! 
I could go for a live performance of Tommy.
I’d love to see a new production of “Amahl and the Night Visitors” on NBC. It could be broadcast from studio 8H at Rockefeller Center, just like the original.
West Side Story is a must–but I’d want them to remember that Tony used to be a tough gang member, and have that come out in his performance (as it did in the best production I’ve yet seen–a small youth company). Richard Beymer played him too much as a squeaky-clean good guy, so it seemed incongruous. (As good as the movie was, they could only go so far in implying certain things in 1961.)
I also vote for Man of La Mancha (and for Brian Stokes Mitchell).
Phantom, in a sense, has already been done–the recorded 25th anniversary performance at the Royal Albert Hall.
It’s arguable whether the concert versions of Les Mis “count” as a live performance, so I’d love to see a new version on live TV.
I would also love to see A Chorus Line or Avenue Q–but their content and language might make them a no-no for network TV. Maybe a cable network could do it?
I’ve often wished HBO would broadcast plays. They used to: they did (the admittedly pretty tame) CAMELOT with Richard Harris back in the '80s and a couple of others, and could rock with edgier stuff.
I saw that version! We actually watched it in class instead of the more well-known movie, after we’d read The Once And Future King. It’s actually a revision of the original playscript–instead of that lame Morgan Le Fay scene, it has the movie’s confrontation between Mordred and Arthur where Mordred convinces Arthur to test his wife’s fidelity (“You wish me to be your son? Then prove to me that virtue can triumph in the absence of the King!”)
They also broadcast a taped version of the William Gillette Sherlock Holmes play. I think Frank Langella was the lead.
Gotta be Springtime for Hitler.
I remember that as well. It actually showed him shooting up with his 7 percent solution.
They also did a one man show about Vincent Van Gogh starring Leonard Nimoy as his brother Theo.
ETA: And all this was when there was just the one HBO and not multiple (including the Spanish one and the one that’s always showing Harry Potter and the Chamber of Secrets).
Looks like NBC will be doing Hairspray. From Cynonpsis, a trade e-zine:
I’d like to see them do Chess.
One more vote for West Side Story. Also: The Music Man with Matthew Broderick was a big disappointment. I would love to see this recast and broadcast live.