It depends a bit on the AVR in my experience.
In general, the HDMI connection goes away when the TV is shut down, and the graphics card in your HTPC stops sending anything. Similar, if your HTPC shuts the display off (ie. idle timer), audio also stops.
With some AVRs however, the connection remains active even with the display is off, as it takes over the connection from the TV.
There probably isn't going to be an easy fix. HDMI in general is only supported rather badly by HTPCs, many use-cases are left out.
Too bad there is no proper general purpose digital audio connection independent of video, so that you could run that to the AVR. SPDIF/optical is bandwidth starved (limited to 2ch or AC3 encoded), and USB is usually not supported by AVRs.
HDMI is the only choice for full audio, but it can leave us wishing for more.