How to Unsave Your Password on Facebook
When you are using Facebook on your personal computer, saving the password so that you stay logged in makes sense. If, however, another person begins sharing your computer, unsaving the password protects your privacy. Both Facebook and your Internet browser can remember your password for the social networking site. You must log out of Facebook to stop the site from remembering your password, but you can modify Internet browser settings even when you are logged in.
Click the "Account" link in the top right corner of any Facebook screen, and select "Logout" from the drop-down menu.
Uncheck the box next to "Keep Me Logged In." The next time you log in, Facebook will not save your name and password to keep you logged in indefinitely.
Clear the saved password in Internet Explorer by going to the "Internet Options" menu and clicking "Delete" in the "Browsing History" section. Click "Delete Passwords."
Go to the "Tools" tab if you use Firefox, and select "Options." Click on the "Security" tab and then "Saved Passwords." Highlight the Facebook line, and click "Remove."