Yes, I know how to fix this. Unfortunately, there also seems to be a bug in the Zimbra Admninistration UI which is preventing the MTA relay to be set. Possibly, another one for bugzilla?
Anyway, to fix using the Zimbra CLI:
SSH to the server and su to the Zimbra user and issue the following command:
zmprov -z ms your.zimbra.server.host zimbraMtaRelayHost the.relay.host:25
1. your.zimbra.server.host is the Zimbra server
2. the.relay.host is the MTA relay host to send email too
You should not need to restart Zimbra (but I do anyway zmcontrol stop && zmcontrol start)
Send some external email and watch the zimbra.log, you should see the email routed externally now via the relay host.
All email should now be sent via the external relay host