Sorry I've been away...
Well, Tweak UI didn't do it. KillTT did indeed kill the tooltips in the taskbar. Sadly, it also killed the tooltips in other programs where I want them to happen.

Ideally, I'd like to be able to set a delay before they pop up, and even more ideally, separately for the task bar and for applications. Oh, well.
Am I the only person out there who's annoyed by the fact -- and thinks it's a little strange -- that these things pop up immediately and obscure the various icons we're trying to work with? In particular the quick launch stuff on the left and the tray icons on the right.
Thanks everyone for trying to help.