When special events occupy a theatre with a show running in it.

first of all, the OP is specifically asking about non-profit organizations. Obviously, a commercial occupant is going to pay retail.

secondly, and I know I sound like a broken record but the nonsense is unrelenting: there are no rules. An organization may pay nothing, if the landlord is so inclined, or may pay the pass throughs. Or may pay more, if the landlord is not particularly charitable to the organization or cause in question.

third, what's in the tenant's license is negotiated, and no one can make a blanket statement as to what the deal is. There are shows where having trespassers is not allowed (for reasons of integrity, if not safety). And there is no agreement that limits a production to the performance schedule. There are a million things that the production may need the theatre without regard to performances (rehearsals, work calls, being the most obvious). At a minimum there would be an advance clearance and depending on the situation, compensation, as there are likely expenses. It would be really cool if people did not post about things they don't know.