I used to have this problem on my computer,
You should be able to modify the permission from the person you would like to stop saving to the desktop by modifying the Desktop folder in Startup Scripts. The Group Policy cannot restrict/add/remove items if they have modify permission.
You can enable the policies by doing this:
User Configuration->Administrative Templates->Windows Components->Internet Explorer->Internet Control Panel,-> Security Page
You can enable "Allow file downloads" policy and select Disable in Internet Zone and Intranet Zone.
