I have the same problem.
Imap and Pop do NOT require ssl or tls
Mta is exim, pop/imap is courier.
I can connect with any other mail client
eg, thunderbird, outlook.
I can telnet to both ports and login.
When I try ZimbraDesktop I get this error:
Cannot connect to "mail.domain.tld:143". Please check host/port and network connectivity.

When I check the zdesktop.log I see: (Domain details replaced for privacy purposes)
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2010-03-31 16:56:58,309 INFO [btpool0-1] [] datasource - Testing: DataSource: { id=2d4b4aff-43c6-4005-985a-9cb1eb1d3c57, type=imap, isEnabled=true, name=user@domain.tld, host=mail.domain.tld, port=143, connectionType=cleartext, username=user@domain.tld, folderId=1 }
2010-03-31 16:56:58,310 INFO [btpool0-1] [] datasource - IMAP read timeout is 60 seconds
2010-03-31 16:56:59,695 DEBUG [btpool0-1] [] imap - S: * OK [CAPABILITY IMAP4rev1 UIDPLUS CHILDREN NAMESPACE THREAD=ORDEREDSUBJECT THREAD=REFERENCES SORT QUOTA IDLE ACL ACL2=UNION STARTTLS] Courier-IMAP ready. Copyright 1998-2005 Double Precision, Inc. See COPYING for distribution information.
2010-03-31 16:56:59,696 DEBUG [btpool0-1] [] imap - C: C01 STARTTLS
2010-03-31 16:57:00,112 DEBUG [btpool0-1] [] imap - S: C01 OK Begin SSL/TLS negotiation now.
2010-03-31 16:57:00,587 WARN [btpool0-1] [] datasource - Test failed: DataSource: { id=2d4b4aff-43c6-4005-985a-9cb1eb1d3c57, type=imap, isEnabled=true, name=user@domain.tld, host=mail.domain.tld, port=143, connectionType=cleartext, username=user@domain.tld, folderId=1 }
com.zimbra.common.service.ServiceException: system failure: Unable to connect to IMAP server: DataSource: { id=2d4b4aff-43c6-4005-985a-9cb1eb1d3c57, type=imap, isEnabled=true, name=user@domain.tld, host=mail.domain.tld, port=143, connectionType=cleartext, username=user@domain.tld, folderId=1 }
ExceptionId:btpool0-1:1270047420586:2445292d93f247fd
Code:service.FAILURE
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