Well this kind of goes against everything I preach but you could modify the base code file to do this. Should be a pretty easy modification, just document it when it comes to upgrade time.
Under /Admin/CMSAdministration.aspx, you'll find there is a section in there for the header, in one of the UILayoutPane controls it references file /CMSAdminControls/UI/Header.ascx. If you look at that Header.ascx user control, you'll see a reference to this control: <cms:ContextHelp runat="server" ID="contextHelp" />. In the code behind on the Page Load method, you could set it's visibility, but I'd suggest setting an attribute to it so "display: none;". I say this because I believe there is some other JS you'd have to worry about if you set it to visibility=false. By setting visibility, none of the code will be there and the JS will error out. By setting an inline style attribute the code will still render the html and not error out any JS that will be there.
Again, not anything I'd recommend but if it needs to be done, then it needs to be done, just be sure to document it.