Even if it was possible to remove or disable "Private Browsing" then that still won't make it impossible to delete private data like visited websites (history) or cache or cookies.
There are a lot of ways to clear such data, either via Delete browsing, search and download history on Firefox or directly by deleting the History in the Library or sidebar.
Private Browsing makes it easier because all data is kept in memory and nothing is saved to disk, so nothing needs to be removed.

If you want to see which sites have been visited then you should look at others means outside Firefox like a router or firewall with a password protected logging feature that can't be bypassed or a program like Wireshark.

• http://www.wireshark.org/download.html
• http://en.wikipedia.org/wiki/Wireshark

See also:

• http://kb.mozillazine.org/Parental_controls

Thanks for your answer.

You might like to work your way through this recent discussion.

https://support.mozilla.com/en-US/questions/827537

This question seems to be coming in every day.

Remember that a new browser of any sort can be downloaded and uninstalled in no time flat. You'll also notice my reference to USB memory sticks.

Cor-el has fully covered how easily you can cover your tracks in ordinary browsing mode. My assessment is that supervision and a password on your operating system are the only answers. You'll notice the reference to "trust" by naive alan_r.

If you are still about you might like to consider these suggestions from Xircal - I know nothing about them.

https://support.mozilla.com/en-US/questions/828454