This sounds like a copyright violation.
You may want to take a look at Castle Rock Entertainment Inc. v. Carol Publishing Group, 150 F.3d 132 (2nd Cir. 1998). That was a case dealing with Seinfeld, where a publisher printed and sold The Seinfeld Aptitude Test, a book of Seinfeld trivia. The publisher argued that the questions and answers were simply unprotected facts about the show, not copyrightable expression, but the court concluded that the facts were merely restatements of the expressive works of the show's creators. Although the book was in some senses transformative, the court found that the transformation was "slight to non-existent":
[T]he fact that The SAT so minimally alters Seinfeld's original expression in this case is further evidence of The SAT's lack of transformative purpose. To be sure, the act of testing trivia about a creative work, in question and answer form, involves some creative expression... However, the work as a whole, drawn directly from the Seinfeld episodes without substantial alteration, is far less transformative than other works we have held not to constitute fair use.
A major part of the problem in that case was that the facts suggested that the publisher was not looking to create an educational or critical work revolving around Seinfeld, but rather to provide Seinfeldesque entertainment to Seinfeld fans.
If I held the copyright to Friends, I would argue that the museum, by trying to merely replicate various scenes and props from the show was merely trying to duplicate the show as a means of piggybacking on my creative expression, and I suspect the courts would be pretty receptive to that argument.
On the other hand, Castle Rock suggests that different behind-the-scenes facts might change the analysis. If the museum it included substantial educational content or critical analysis of the show, it might fare better than the Seinfeld trivia book.
As a practical matter, I'd also note that regardless of whether the museum could correctly be considered a legal fair use, the copyright holders would almost certainly try to shut it down or extract licensing fees anyway. If I were not prepared to litigate the issue, I doubt I'd be willing to put any money into launching this operation.