Thanks for the replies, sorry it's taken a few days to get back online - work schedule has been heavy.
I've looked in the Playlists folder as well as the Library folder of my MusicBee files, and nothing is marked as Read Only.
I went through my Firewall Settings and MusicBee wasn't whitelisted for some reason. I have it whitelisted now but that didn't fix the problem.
For reproducing the error, I'm just opening MusicBee, going to a playlist, and clicking on one of the sort options in the header bar, then the error message pops up. As far as I can tell, everything else is still working fine - music still plays fine, there aren't any playback issues, and I can edit the song info without a problem. It only comes up with this error message when I click "artist", "album", "year" (or etc.) to sort the playlist. The only thing that I can think of that is different now from when MusicBee was working fine, is in backing up the playcounts and ratings from iTunes after synching my phone... But it backed up fine and didn't have any issues in doing so.
EDIT: I closed MusicBee after writing this comment, and got an error message that it was unable to save changes to the library because access was denied. So it does sound like there's an access issue somewhere but if nothing has changed itself to Read Only then any suggestions where it could be? I tried opening MusicBee as Administrator and that didn't change anything either.