How do I get a clean full screen hdmi output from the Nikon D800/D800E? 

This is not very clear in the Nikon D800/D800E manual, but I've managed to find the trick: you have to turn the 'live view on-screen display' menu option to 'off'.  Location in the menu: wrench, hdmi, advanced (see picture).  This works great and give full screen output through hdmi, but surprisingly the display on the back of the Nikon D800/D800E still stays on during live view even though the setting suggests otherwise. 

This trick only works for live view and video playback.  For viewing photos there is no setting to show a full screen picture without the menu clutter left and right, unfortunately. 

To prevent clipping in the white and dark areas, most external devices require the 'Output range' (wrench, hdmi, advanced) set to 'limited'.  This includes some monitors en professional external hdmi recorders.