I have the HDMI adapter and it works fine with the display off.
I use it when I travel. I log into my Sling Box and hook it to the hotel TV's that have HDMI inputs. It's great because I get all my DirecTV stations. Also makes great use of HBOGO, MaxGo, Netflix etc. One thing I also carry is a programmable remote control so I can access the TV's alternate inputs. Sometimes you have to disconnect the cables from the hotel's video network in order to access the alternate inputs but that is not a big deal.