That error seems to come from your SMTP server and means that the server encountered a network problem when trying to connect to the target server.
Do you need a DNS server? Well, yes but no, that server must not necessarily be on a particular machine. One well proven approach would be to have a DNS "relay" (correctly, a forwarding server) like unbound on your gateway system (I think it's 192.168.1.1 in your case). But - and careful, that's important! - every host in your network incl. your mail server needs to know (and have access to) a DNS server.
So, concrete, for your situation:
P.S. The error you mentioned seems to indicate a network problem rather than a DNS problem.
Do you need a DNS server? Well, yes but no, that server must not necessarily be on a particular machine. One well proven approach would be to have a DNS "relay" (correctly, a forwarding server) like unbound on your gateway system (I think it's 192.168.1.1 in your case). But - and careful, that's important! - every host in your network incl. your mail server needs to know (and have access to) a DNS server.
So, concrete, for your situation:
- Be sure to have some DNS server available and reachable, either on your gateway or some external server like you ISPs DNS server.
- Be sure that every host in your network knows about that DNS server (you can set it in /etc/resolv.conf)
host [URL='http://www.google.com']www.google.com[/URL]
P.S. The error you mentioned seems to indicate a network problem rather than a DNS problem.