This message pops up after a few minutes of disuse:
"You are currently offline. Do you want to go online to get new messages?"
Sometimes the magic that detects whether your offline is faulty.
To disable it, open Thunderbird's configuration window (Edit > Preferences or Tools > Options depending on your platform), go to the Advanced panel, select the General tab, and click Config Editor. In the window that appears, type toolkit.networkmanager.disable
in the filter box at the top. Double-click on that option to toggle it to true.