I have a similar problem with a Zenbook Prime UX31A. I tried 3 different cables and 3 different displays and nothing works. I can't even get a picture in the BIOS, so I tried all available BIOS versions. The port seems completely dead to me, the displays shows absolutely no reaction when I plug it in. I checked the +5V line and voltage is present.
I'm about to say this is a hardware related problem. I haven't been using the HDMI port for some time, so I can't remember an event with which the problem started. But it was working before.
VGA is working by the way. Perhaps some users have the same problem as I have. Is there someone without picture while booting?
I'm out of ideas and appreceate any help.
UPDATE: I am able to activate HDMI in Linux manually. So it seems to be a hot-plug or edid related problem.