This problem could be produced by different causes:
1) Wrong permissions on the Zimbra files/directories, shutdown Zimbra and run the following (as root):
restart Zimbra and see if that fixes it.
chown -R zimbra:zimbra /opt/zimbra
2) If the problem persists, do the following:
and post here the output.
dig yourdomain.com mx
dig yourdomain.com any
3) The certificates on the server are expired
you can find the expiry date in the 'notAfter' attribute.
/opt/zimbra/bin/zmcertmgr viewdeployedcrt all