As it turns out, a Fantastic Four Broadway musical is possible, so long as you have the licenses in place, actual talent, and it's based on real events as they do in the Marvel Universe. Arrested Development's Tobias Fünke had none of this, and still tried to get his musical about Marvel's First Family off the ground anyways, which was a hilarious disaster in the popular show's fourth season from Ron Howard. In Spider-Man #25, Mary Jane Watson ends up saving the show from the new Electro, gaining herself a round of applause.
In the fourth season of Arrested Development, Tobias Fünke is still trying to become the famous actor he's always dreamed of being, hoping to ride the perceived "coattails" of his new girlfriend DeBrie Bardeaux, a recovering methadone addict and former actress. Her acting career involved being quickly cast as the Invisible Woman in a haphazard production of Fantastic Four as a way for Imagine Entertainment to simply retain the rights to the IP. This mirrors the actual 1994 low-budget film from Bernd Eichinger and Roger Corman that was made for the same reason. At the time, DeBrie was simply a caterer at Imagine's Christmas party.
Together, Tobias and DeBrie try to make money by dressing up as The Invisible Woman and The Thing on the street, though they are frequently sued by Feinberg, Feinberg, Feinberg, and Feinberg for copyright infringement. Despite them both ending up in rehab, Tobias persists, convincing Debrie, several of the patients, and the director of the rehab clinic to put on Fastastic 4: An Action Musical. Together, they tried to perform during the Cinco De Quatro festivities on the water, a poorly thought out attempt to evade the Marvel lawyers.
As previously mentioned, a Fantastic Four musical apparently is a lot easier to do well in the Marvel Universe. In Spider-Man #25, written by Nick Spencer with art by Ryan Ottley, Mary Jane Watson and her friend Carlie Cooper go to see "Galactus", a broadway musical depicting the adventures of the Fantastic Four. Galactus himself is a big part of the set as a giant animatronic puppet (akin to the King Kong animatronic on Broadway in real life). Also, the Invisible Woman is played by Melanie Daniels, an old acting rival of Mary Jane's from when she was still acting in Hollywood. However, midway through the performance, the new Electro attacks, taking Daniels hostage while live-streaming the whole ordeal. Using some quick thinking, Mary Jane and Carlie end up saving the day and defeating Electro with a giant tank of water, all without the assistance of Spider-Man. When the dust settles, Mary Jane gets a standing ovation for her heroics and even gets some offers to return to acting (though one of those calls is to star in a film secretly produced by Mysterio.)
While this particular performance of the Fantastic Four musical was interrupted, it's implied that it had been an ongoing musical on Broadway with a certain amount of success in the Marvel Universe (something that could not be said for Tobias' version.) It does beg the question though: Do the actual Fantastic Four get royalties for their life stories being played out in song? How does that work? It seems like there would have to be some sort of license contract in place. In any case, it's pretty fun to see what the performing arts look like in the Marvel Universe all the same, if only a small part.