This line seems to be the culprit. (from /var/log/zimbra.log) When I view the queue in the admin, there are 16 deferred messages all indicating the same error.

Jun 29 08:00:30 mailbox postfix/qmgr[4969]: 2A775D5872D: to=<>, relay=none, delay=0, status=deferred (delivery temporarily suspended: Host or domain name not found. Name service error for type=A: Host not found, try again)