Firefox 20 added new feature, that allow you to open private browsing window in same browsing session.

I don’t want this new context menu. To hide this, go to your firefox profile. You can find this by going to

FireFox > Help > Troubleshooting Information

You can directly go to Troubleshooting Information page by typing about:support in address bar of firefox.

On next page, click show folder.

In your profile folder go to chrome folder.

Copy userChrome-example.css to userChrome.css

Now add

#context-openlinkprivate { display:none!important; }

To the file and save. Restart firefox. Now the link will be gone.

I had already know how to hide Open In New Window was context-openlink, so to find for Open Links in a New Private window, i downloaded firefox 20 source code and grep for context-openlink.

[root@server12 mozilla-release]# find ./ -name '*.js' -exec grep 'context-openlink' {} \; -print
    this.showItem("context-openlink", shouldShow && !isWindowPrivate);
    this.showItem("context-openlinkprivate", shouldShow);
    this.showItem("context-openlink", shouldShow);
    this.showItem("context-openlinkintab", shouldShow);
    this.showItem("context-openlinkincurrent", this.onPlainTextLink);
./browser/base/content/nsContextMenu.js
  let linkMenuItem = document.getElementById("context-openlinkincurrent");
./browser/base/content/test/browser_plainTextLinks.js

[root@server12 mozilla-release]#

Here is the content of my userChrome.css, i have disabled Open in New Window.

@namespace url("http://www.mozilla.org/keymaster/gatekeeper/there.is.only.xul");
#context-openlink {display: none !important;}
#bookmarksToolbarFolderMenu { display: none !important;}
#organizeBookmarksSeparator { display: none !important;}
#context-openlinkprivate { display:none!important; }