If a player shows up in Chamber of Lost Souls, what are my responsibilities as GM of an adult game they are in? Do I have to delete them from my game?

By definition if they are there they are removed from the site right? NPC their character maybe?

No, you are not required to delete them from your game, they will simply not be able to access it until whatever the issue was with their account is resolved.

It's always good to pay attention to the Chamber, tho, as that may be the only way you can tell the person is banned and not just not posting.

And I don't know about anyone else, I much rather remove the person and they can RTJ again once things are resolved. Though if it has to do with the more adult part of things, I probably don't want them in my game anyways.