The easiest way may be to set up the account as POP and disable all checking for mail at startup and periodically. IMAP accounts might still 'push' notifications, even if the settings in TB disallow them, depending on the mail service. But setting up an 'unimportant' account as POP is probably not a disadvantage.

Thanks for the possible workaround. Setting up as a POP would not be workable, however. I'm going to asume that you offered a workaround because, the feature I'm requesting is not currently supported. If that's the case I would have to submit it as a feature request (..and perhaps ask why something so mundane isn't already a feature by now.)

There are a number of similar requests on Bugzilla:

https://bugzilla.mozilla.org/show_bug.cgi?id=988504

It's probably not a high priority because most users have accounts in TB for the purpose of receiving mail. You could add unimportant accounts to a separate profile and launch it when necessary.