There are no "rules" that I am aware of. If you are nominated and invited to perform, then it is your option (not obligation) to accept or reject the opportunity to showcase a performance from your production.
In years past, there have been performances by shows which had already closed (ie, SIDE SHOW).
It wouldn't make much sense for a production to perform at the Tonys if they have already closed. It costs money to pay the actors, designers, director, choreographer, etc. You have to construct a new set for the performance (shows cannot remove their set pieces from the theatre, but they can move costumes and props), too, which also costs money.
It WOULD make sense for a closed production to perform, however, if that production were to go on National Tour.
"The Spectacle has, indeed, an emotional attraction of its own, but, of all the parts, it is the least artistic, and connected least with the art of poetry. For the power of Tragedy, we may be sure, is felt even apart from representation and actors. Besides, the production of spectacular effects depends more on the art of the stage machinist than on that of the poet."
--Aristotle