Just to clarify: the primary password protects the email accounts themselves; it is not the actual account password. Use of primary passwords is optional and is a PC security feature. Removing the primary password requires deleting (or renaming) some files in your profile (key4.db, logins.json and cert9.db), as there is no way to delete it in Thunderbird. You will then need to re-enter the email account password. If you do not know the email password, you can contact your email provider for assistance in resetting it.

Open the Error Console (Ctrl+Shift+J), paste this in the bottom box:

chrome://pippki/content/resetpassword.xhtml

press Enter, then Reset. See the warnings in the article below:

https://support.mozilla.org/en-US/kb/reset-your-primary-password-if-youve-forgotten-it

In the meantime, you should be able to access new mail through your provider's webmail.