I have found a very short play by a well-known playright and I would like to seek permission to get festival-only rights for making it into a short movie. I realize they are likely to just say no.

However my question is this: should I ask the playwright's agent, the publisher of the book the play appeared in, or someone else? I haven't worked with play scripts before.

Contact the agent first or if you have the means, try to contact the author. It's been my experience that writers are usually more then happy to have their work showcased if it remains true to the story. I've optioned several short stories and scripts, while there's not a huge market for them, one still does exist, especially the festival route. With a feature all parties involved really anticipate a return on their investment, with a short everybody knows it's a "calling card". Any exposure to an authors words is better then none at all.