PbP Campaign to Audio Play... Legal Ramifications?
The first thing I would recommend is contacting Paizo to ask them for permission to adapt or incorporate their material into a radio/audio-play. Different companies have very different approaches to how protective they are of their intellectual properties. Lucasfilm, for instance, apparently doesn't care all that much about fan-films, as long as nobody's making money on them. Paramount has released a ridiculous list of restrictions for anyone wanting to make Star Trek fan films, but they'll still allow it (they just can't be any sort of ongoing storyline...and no single episode can be more than...I think it was fifteen minutes in length? And you can only use uniforms and props from authorized vendors, and...you get the idea).
They may love the idea, and tell you to run with it (I mean, there are other episodic publications out there doing D&D in various forms, so I personally think it would be a good thing for them to get some broader exposure in the same vein). They may tell you that you absolutely cannot use any of their intellectual properties for any sort of self-produced work. They may want to see the script and decide that they actually want to officially produce it. You never know...
If they say 'no', you can always change names and keep the story the same, just make everything generic enough to be any fantasy world, or make your own world and change names to be specific to it...just change it enough that it's no longer a Pathfinder setting (it can be explained as being inspired by Pathfinder...there's still a chance that it could come under legal action as being a derivative work, but that would require a lot of scrutiny from their legal department before they jumped in on it and if you make it different enough, you've got good chances of winning any case they want to try and make).
And they may say nothing at all...which is, in an of itself, a limited amount of legal protection for you. I'm not sure where the law stands now, but twenty years ago when I was looking at such legalities, a show of good faith on your part to gain permission, which gets no response, equates to implied permission.
I would also strongly suggest getting some kind of permission or contractual arrangement with all the players, because the argument can be made by them that, in using their words, character, etc, you are infringing on their intellectual property. It would be hazy, because I don't know that any sort of copyright law has been written for this kind of situation...but if you cover the bases beforehand, you don't need to worry about that. If your game is flagged sole-ownership, there's an argument to be made for not needing that...but I'd suggest treating it as much as possible as a group writing project and getting permission from everyone who contributed to use their work, just because it's ethically sound, even if it may not necessarily be legally necessary. And of the two considerations, I'd say this one is the greater one to worry about. It's easy enough to change details and omit bits and pieces that would make this specifically a Paizo-owned world, and there's so much fantasy literature and entertainment out there that they'd have to be REALLY desperate to try and make the case that your name-changed world is the same as the one they published. But if they give you the okay, you don't have to change names, and you can also potentially use the name-recognition value of saying you're doing a Pathfinder story to help promote your work. But getting a few months into something and getting hammered by bad press and potentially lawsuits over players who feel their contribution to the work has been misrepresented as being someone else's work can do a number on you, in a hurry.