You can autohide tabbar in Thunderbird:

1. Open Thunderbird’s Preferences window (Tools -> Options on Windows, Edit -> Preferences on Linux and Thunderbird -> Preferences on Mac)
2. Click on “Advanced”
3. Click on the “Config Editor” button (which is under the “General” tab)
4. Search for “mail.tabs.autoHide”
5. Double-click on the setting so that it becomes “true” (and the font changes to bold)

Voilà!

If You want to permanently hide it use this code.