Update: This solution does not seem to work anymore in many browsers. Kaitsu’s comment:
This solution of sending false credentials to make browser forget the correct authenticated credentials doesn’t work in Chrome (16) and IE (9). Works in Firefox (9).
Actually you can implement a workaround by sending false credentials to the service. This works in Browsers by sending another (non-existent?) Username without a password. The Browser loses the information about the authenticated credentials.
https://www.example.com/ => Log in
with basic auth as “user1”
You’re Logged out. 😉
P.s.: But please test this with all needed Browsers before you rely on the given information.