I've been using the following setup:
Gigabyte GA-H81M-HD3 + Celeron G1820 -- HDMI --> Onkyo TX-SR576 -- HDMI --> Panasonic TH-42PZ85E
With the lastest Intel drivers, 10.18.10.3412, and the latest _OFFICIAL_ BIOS (F3, released 24-jan-2014) I got all of the problems listed in this thread. Turn off receiver --> lost audio, change input source on receiveer --> lost audio.
Only way to get back the HDMI audio device was to reboot the PC.
I got in contact with gigabyte and notified them about the problems and the potential BIOS update. On the 13th of march I got the F4a BIOS for my motherboard, it's dated 7-mar-2014, so definately newer than the official F3 BIOS.
With the new F4a BIOS everything is working perfectly: I can turn off my receiver or switch sources on my receiver and the HDMI audio keeps working!
For everyone else with the Gigabyte GA-H81M-HD3 motherboard, here's a dropbox link with the F4a bios: Dropbox - H81MHD3-F4A.zip