I opened a support ticket and the problem has been resolved thanks to LunarPages technical support.
My server (HEKA) was migrated to Centos 7, which no longer supports suPHP. My .htaccess file (at the root and in a couple of my domain folders) had some suPHP_ConfigPath lines in them. E.g.,
Commenting these out got things working again. So for those of you dealing with this problem, check your .htaccess files, both at the root and in your domain directories. I don't know why I had this line in there, but it was probably added years ago to fix some other problem.