It turns out, that the problem had to do with mod_autoindex. The directory in question is setup to use mod_autoindex and list a bunch of changelogs in the directory. For whatever reason, mod_autoindex wants to access .htaccess while it's building a list of files and since .htaccess files are disallowed for all in the normal Apache configuration, it's getting that permission error.
I've found that adding .htaccess to the IndexIgnore directive will stop this error from happening. Since I also have mod_autoindex ignore the link to a parent directory, my IndexIgnore directive looks like this.
IndexIgnore .. .htaccess