Session_start(): ps_files_cleanup_dir: opendir(/var/lib/php5) failed: Permission denied (13)

A great article has been published on this error particularly. In summary, the error is caused by PHP is trying to delete the session files while this is also being done by a cronjob put on the system by the Debian package maintainers of PHP.

The easiest solution is changing the permissions which is typically www-data:www-data.

chown -R www-data.www-data /var/lib/php5