How did you connect to your company's Exchange server? Unfortunately, Thunderbird does not include Exchange ActiveSync (for license reasons) so you probably used either IMAP or POP. If you used POP, what you see in Thunderbird is just a copy and your local changes have no effect on the server. If you used IMAP, then the changes you make locally should sync up to the server.

I'm using IMAP, but as mentioned, if I drag an email from my inbox to a folder in Thunderbird, it remains in the inbox in Outlook.

I'm a little confused by this article: http://kb.mozillazine.org/Deleting_messages_in_IMAP_accounts

Another discussion for reference: http://superuser.com/questions/452740/how-can-i-make-thunderbird-imap-sync-moved-messages-immediately

The second link seems to describe my problem (or close to it), but the "solution" suggested (link 1) doesn't seem to help.

Lets see. OWA is not IMAP. You can access exchange mail via OWA. Either using davmail or the exequilla add-on.

The IMAP implementation in exchange is odd. This is mainly because Microsoft never intended anyone to actually use it, but marketing needed to be able to say it supported it. So have you set Mail.imap.expunge_after_delete to true and restarted Thunderbird.

Many settings that have no user interface are only read on startup of the program.

Thanks much, Matt. Bizarre to me (as I don't really understand these things), closing Thunderbird and restarting solved it.