NOWaWarning said: "Just out of curiosity, does anyone know how much a show has to pay to perform?"
I may be 100% wrong, but my understanding is that it's not the Tonys charge a huge rate to do it (I don't think they charge at all, but could be 100% wrong again), it's that shows have to pay for everything associated with it.
The Hadestown clip below is a great example where they probably had to pay someone to write that whole opening section, give the director and choreographer billed hours to stage and choreograph that section and make the adjustments needed for the new stage without the turntables, several sections of the song rempoved, and with all the Swings in it (Hadestown usually only has 5 ensemble members), pay for a rehearsal space, pay all the actors and musicians for rehearsal time to learn all of that, and transport the costumes and props to Radio City. Plus, I'm not sure about this one but, I wouldn't be surprised if they made them pay for and build the new set and to bring their own har, makeup, and costume people.
https://www.youtube.com/watch?v=MWtjGIV1sMQ