With regular Gmail accounts, and I presume also with Business accounts, a copy of the sent message is automatically stored in the IMAP Sent folder, so you should uncheck "Place a copy in" under Tools/Account Settings/accountname/Copies & Folders. Does this make any difference?

Ah! Fantastic. That solves the issue for three of my accounts.

The remaining mail account is on HostGator. Since Gmail's spam filter is massively better than HostGator's, I've got it set up to forward all mail to a Gmail account, which I then check in Thunderbird and reply to using HostGator's SMTP so that my outbound mail goes from the proper address. Would the same technique apply with a non Gmail SMTP?

Yes, any service that automatically copies sent messages will work the same. That's the case with Gmail and Hotmail/outlook.com, but not all services. A simple test mail should confirm the status of HostGator.

Awesomesauce. Thanks!

I don't see the point of the above answer,

I'm using Thunderbird with a Googlemail account so that I don't need to look at the web-based interface for Googlemail.

If I uncheck Save copies, it won't save a copy at all - that would be disastrous.

Why does Thunderbird sometimes refuse to save a copy ?

How do I resolve this ?

You can easily send a test message and see if sent messages are automatically saved, even with the option turned off in TB. If you have chosen to synchronize the Gmail Sent folder, that folder should show the sent messages in TB. That's how it works in Hotmail/live/outlook.com accounts, and it should be the same with Gmail.