If you use chrome, you can use an incognito window (ctrl+shift+n). You can also disable third-party cookies (just remember to delete current facebook cookies). Since iOS devices block third party cookies, most sites that wouldn't work with them enabled now do. If you use chrome, you can also disable them from being read (in about:flags). Chrome will also show an icon in the urlbar if a site wants to set third party cookies, so you can enable them for specific sites (it also has the same behaviour for blocked javascript and plugins).

The other option is to totally disable platform apps, in the privacy, applications menu. This will mean you can never log into an external site, but it also means that you can't have any applications. I would do it myself if I didn't use a third-party android application, that requires a facebook app for interaction