Beaufoy, who scripted the hit 1997 film and the recent stage adaptation, which premiered at Sheffield Theatres last year, has added his name to an online petition to keep the show running until June, when it was originally scheduled to finish.
Beaufoy writes that he is "devastated" the show is closing early, and "pondering how on earth that can happen to a play that regularly gets standing ovations".
"If it was a bad show, I'd understand- or even a ho-hum sort of affair. But read what the Telegraph and the Times have to say about it, or better still read on Twitter what the audiences felt about it. On Monday, we got an Olivier nomination for Best New Comedy: on Saturday we were given notice. That's a steep dive by anyone's standards."
He then quotes an email from a Delfont Mackintosh employee to a ticket holder, which implies the decision to close the show was taken by the producers rather than the theatre owner.
Beaufoy adds: "[this email] begs the question why would a producer close his own show before it had any chance of finding an audience? On the other hand why would a producer never come to see his own show in the West End?
"Perhaps he can answer these questions. I can't because he hasn't spoken either to me or the creative team in months. He absented himself utterly from the West End run of the play. Without his undoubted skills and experience, a producerless show faltered."
The petition on Change.org has already attracted over 700 signatures.
The news of the closure was broken by cast member Kenny Doughty on Sunday, when he revealed the cast had been told after the Saturday evening's performance that the show would close.
Producers David Pugh and Dafydd Rogers commented on Monday: "It is with great sadness that this show will close despite having played to packed houses on its UK tour. Simply not enough of a West End audience shared in the excitement of this terrific production."
The Full Monty is due to recommence its national tour with a new cast in September.