The latest Intel HD driver with the latest motherboard BIOS fixed it for me.
I had the issue with HDMI audio working fine on bootup, but going away after the monitor/TV was turned off (requiring a reboot to get it back).
I already had the latest Intel HD drivers when I discovered the problem. Reading through this whole thread (egads, an hour of time there), I decided to try the latest MB BIOS from MSI (it didn't mention Intel HD audio, but said "Updated VBIOS and GOP Driver", perhaps that was it). Voila, problem solved.
So at least in some cases, a MB BIOS update can get rid of the problem (with the latest Intel HD driver).
-Scott