It seems that the domain doesn't have any MX records at all.
This is the response from looking up the Zimbra MX records, and how properly-configured MX records should look like.
Whenever messages are sent to @zimbra.com, mta01.zimbra.com will receive them and deliver them accordingly.
;; QUESTION SECTION:
;zimbra.com. IN MX
;; ANSWER SECTION:
zimbra.com. 1200 IN MX 5 mta01.zimbra.com.
zimbra.com. 1200 IN MX 15 dogfood.zimbra.com.
As your domain does not have any MX records (servers that will handle the email), anybody sending messages to people at your domain will not get delivered as essentially (as far as DNS is concerned) there is no mail handler set up.