I started getting this error a while back and have several times looked for a solution without success.
Secure connection failed
live.mozillamessaging.com uses an invalid security certificate.
Now, in the first instance this shouldn't really occur. Whose fault it is I don't know. But they need a kick up the arse.
I have just ignored it and carried on regardless as it hasn't affected me other than being annoying.
But I worked out a very simple fix.
This is in Thunderbird 17 under Linux but guess it will work equally well on other versions.
Edit - Preferences - Advanced - Certificates
Click View Certificates
Then select the Servers tab
Now Add Exceptions and type in https://live.mozillamessaging.com
Click Get Certificate, View it if it means anything to you, and then Confirm Security Exception.
Job done and no more warnings - I guess it is because of a naming error between www.mozilla.com and live.mozillamessaging.com
Hope that helps