For historical reasons, I don't want to remove a couple of old e-mail accounts in Thunderbird. For one account, TB tries to connect once, and after I blip off the error message, it lets me go about my day.
For the second account, however, the new release of TB annoys me every few minutes with a connection failure message. I'm currently working around this by right-clicking the title bar of the error/signon window and minimizing it (even though it has no minimize button per se) and letting the minimized window stew itself on my task bar.