Failed to obtain lock on file /var/run/nagios.pid: Permission denied

This error is caused by Nagios not having access to the folder where it should save it`s PID file. You can change the path for the PID file (and then also fixing the issue) by editing your nagios.conf (usually /etc/nagios/nagios.conf).

Locate the lock_file setting:

lock_file=/var/run

And change it to another path, like for example /tmp.

Start up Nagios again and it should be much happier