You simply delete the password Thunderbird has remembered. http://kb.mozillazine.org/Setting_and_changing_email_passwords

I deleted the server password and followed that link to see what it said. Unfortunately, I'm not getting prompted for the correct password, as the article said I would be.

Now what?

Sure you deleted the right one? You will typically have two logins, one for incoming (POP or IMAP) and another for outgoing (SMTP). You probably need to delete them both.

I deleted both of them. :-(

I came across a similar case not so long back, where someone had removed both mailbox and smtp passwords. Then they restarted Thunderbird, but were not prompted for a password.

In that case, it was discovered that their Anti-Virus program was causing the problem. I think they were using AVAST but cannot be certain.

They stopped the Anti-virus from running. Restarted Thunderbird and got the password prompt. So please check this out and report back on results.

I tried closing and re-starting Thunderbird yesterday, before you sent this. Now I'm getting the password prompt, but when I enter the server password, Thunderbird still isn't letting me log in and get emails.

It's pretty frustrating--I do use Avast, though.

I suggest you try the following, restart the operating system in safe mode with Networking. This loads only the very basics needed to start your computer while enabling an Internet connection. Click on your operating system for instructions on how to start in safe mode: Windows 8, Windows 7, Windows Vista, Windows XP, OSX If safe mode for the operating system fixes the issue, there's is an problem, almost certainly with avast, blocking your attempts to get mail.