One thing that bugged me about Thunderbird was that your local network IP was always sent with your e-mail. I can’t think of any good reason to do this.
If you look at the header from an e-mail sent from Thunderbird you will notice something like this.
Received: from 192.168.1.100 ([1.2.3.4]) by your.mailhost.net
Where 192.168.1.100 would be your local network IP and 1.2.3.4 would be your internet IP.
We can’t do much about the internet IP. That’s added on by the mail server. But we can at least remove the local LAN IP.

Change Local IP in Thunderbird Email Header
To change what Thunderbird attaches to the e-mail header we need to go into the config page. The location seems to change a little based on your OS and the version you have. But for me right now I can find it by going to “Edit > Settings” then scrolling down and clicking the “Config Editor” button.

Then you want to search for “mail.smtpserver.default.hello_argument”. If you can’t find it then make it as a string option.

Then type what Thunderbird should say instead of the IP address. I just put localhost.
Now when you look at the header you will see something like this.
Received: from localhost ([1.2.3.4]) by your.mailhost.net
A Little More Privacy
I wish we could hide the internet IP just as easy but at least now your not leaking information about your local network in the Thunderbird Email Header every time you send an e-mail.
A Possible Way To Hide Your Internet IP?
The only way I know to do this is to use a mail server that will not post it in the header. Or to use web mail. Most web mail clients I have seen show the IP of the server the web mail client is hosted on. So that might be a work around to check into.
But if your using something like gmail then you might as well not waste your time trying to keep any privacy.