In case anyone else has this problem, avast support sent me here.

http://www.avast.com/faq.php?article=AVKB91#artTitle

The instructions are pretty good, but I’ll mention two things I did differently.
If you already have SSL scanning enabled in avast, uncheck it, delete any individual entries for your email accounts, and re-enable it. That’s the first step.
Also, the directions for the menu items within Thunderbird are not quite spot-on, but they are close enough you should be able to find the right selections. Delete any existing avast mail certificate in the list before you import the new one, otherwise it will just tell you it already exists.

Good luck!