Open whitelisted sites in private tabs automatically. Requires Infocatcher's https://addons.mozilla.org/en-US/firefox/addon/private-tab extension for Firefox. If you lost your whitelist when the script updated, check the script homepage.
Open whitelisted sites in private tabs automatically. Requires Infocatcher's Private Tab extension. This is mainly aimed at freshly opened tabs and an attempt is made to close the original non-private tab.
In about:config, set dom.allow_scripts_to_close_windows to true to allow closing non-private tab.
Remember to @match/@include some sites to make the script work for you.
Come back here when the script updates and overwrites your whitelist!