To build on the earlier messages, it has recently become pretty common for Best Play to close fairly soon after the Tonys. Before that, I think many Best Plays tried to have replacement casts, although The Coast of Utopia closed about a month before the Tonys in 2007. Now, there's a mixture of replacement casts in some cases, and the show closing when the original cast is done in some cases.
Red closed two weeks after the Tonys in 2010. It did not extend after it won, although I seem to recall that it recouped during that last week of performances. All the Way in 2014 also closed in June, 3 weeks after the Tonys. Oslo in 2015 and The Ferryman in 2019 closed in July. Vanya and Sonia got to the end of August. The rest since 2010 (War Horse, Clybourne Park, Curious Incident, The Humans, and Harry Potter) all made it to at least the fall.
So half of the winners since 2010 were already closed at the time of the Tony Awards or within about a month of the ceremony. That seems likely to occur this year as well. Leopoldstadt and Fat Ham are the only nominees still running. I don't think Leopoldstadt will extend past its current closing date of July 2 even if it wins as expected. Fat Ham is currently scheduled to close on June 25. It may extend by a week or two if it won, but probably not more than that.