How do I permanently delete my Deleted Messages folder?

Is this an IMAP account? If so it's likely that the folder exists on your email provider's server, so when you delete it from Thunderbird it is reinstated. To remove it from Thunderbird you will have to unsubscribe the folder first. Right click on the Inbox folder for that account and select 'synchronise'. Select the Deleted folder in the list and click 'unsubscribe'. Then click 'OK'. At that point I usually close and restart Thunderbird to ensure the change has taken effect. (Note that when you look at the list of subscribed folders you may not see the Deleted folder listed. In my accounts, anyway, the folder that is named 'deleted' on my computer is called 'trash' on the server. It seems that computers only understand American!)