How do I set Thunderbird to notify me of new emails in only certain folders?

1. Right click on the folder you don't want notifications for;
2. Select "Properties";
3. Uncheck the checkbox "When getting new messages for this account, always check this folder".

2021: Answer now seems to be the Mailbox Alert extension, which allows setting of alerts down to folder level and works in current Thunderbird version as of this message 78.6

Maybe it is possible with filters and the FiltaQuilla Add-on? The add-on adds many new filter actions and one of them is to disable notifications. You could try to add a new filter rule without conditions, action “Do not notify” and then apply it to the All Mail folder. (I haven’t tried it myself.)