The Band's Visit... watermelon. You could easily make it a drink or salad if that's a more convenient way to eat it. I forget what Katrina Lenk and Tony Shalhoub have when they go out to eat but there are plenty of eggplant, chickpea, and kebab dishes that would be good party foods or sides.
My Fair Lady... tea, possibly alcoholic or iced for a nighttime party. Or you could just serve drinks in tea cups. I also like the idea of cookies or desserts decorated with flowers if you don't want to make some kind of floral salad.
Frozen... sandwiches. It's also a cheap excuse for frozen drinks.
Mean Girls... cheese fries. Also, anything pink.
Spongebob... sponge cake. Star-shaped cookies. Calamari (sorry, Squidward).
Farinelli... you could have some Spanish dishes and tapas or maybe little opera cakes.
Harry Potter... just go to Pinterest. It's really easy to find tons of things with edible and non-edible nests. Also, anything stick-like can be a wand. Cheese straws, chocolate (if you've got a mold), celery, etc. And I don't think they're in the play but you can always go with the old standards (butterbeer, chocolate frogs, jelly beans, etc.).
Junk... cheap excuse for Junk food. I think cookies are mentioned at the end but no one is going to get that.
Latin History... I don't know if food is mentioned in the play but this one opens up a lot of possibilities.
Once On This Island... I would go for a mango or plantain dish since they're specifically mentioned. You could also do something creative to represent the tree. And there are various hurricane and "stormy" drinks.
Carousel... I don't know if you want to deal with seafood but there are some light fish dishes that could keep well. I think you'd be safer going with general fair food (popcorn, cotton candy, etc.).
Summer... pixy stix.
Children of a Lesser God. I forget what they get at dinner. I know it's not the veal. But some kind of Italian food.