Fixed it. Looks like somebody else here managed to disable suexec.

I have suddenly started getting this error in my PunBB installation
when trying to edit any settings from the admin panel:

PunBB reported: Unable to write quickjump cache file to cache directory. Please make sure PHP has write access to the directory 'cache'

I have seen various "solutions" online suggesting changing the
cache directory permissions to 777, but I know this is not required
in my case as I am using suexec and also have a nearly-identical,
working setup on a remote server with the cache directory permissions
at 755.

Does anybody know how I can track this problem down?