ANSWER:

Some background in http://kb.mozillazine.org/Deleting_messages_in_IMAP_accounts.

It is perfectly appropriate for a deleted message in a IMAP account to be visible in webmail or other email clients. Deleting a message just sets a flag, and it’s up to the GUI what it should do when there is a message with that flag set. However, once Thunderbird compacts (expunges) that folder, the deleted message shouldn’t be visible any more from webmail or any email client because it has been physically deleted.

You can set Dovecot to notify clients like Thunderbird that there have been changes to the folder. I found a description of the solution here:
http://www.dovecot.org/list/dovecot/2011-August/060812.html

They say that the CONDSTORE capability is the problem. You should remove CONDSTORE (and also QRESYNC, since it implies CONDSTORE).

Directions:

$ telnet localhost 143
telnet> a login user pass
telnet> b capability

ctrl-] q to get out of telnet session

Take the string you got, remove CONDSTORE and QRESYNC, set “imap_capability” to this value in /etc/dovecot/dovecot.conf.

Mine ended up being this:

imap_capability = CAPABILITY IMAP4rev1 LITERAL+ SASL-IR LOGIN-REFERRALS ID ENABLE IDLE SORT SORT=DISPLAY THREAD=REFERENCES THREAD=REFS MULTIAPPEND UNSELECT CHILDREN NAMESPACE UIDPLUS LIST-EXTENDED I18NLEVEL=1 ESEARCH ESORT SEARCHRES WITHIN CONTEXT=SEARCH LIST-STATUS

by Alan Porter from http://superuser.com/questions/422985