[sspectre]

Sadly, the cheap HDMI switch I bought didn't work (for my purpose).
It was a "Portta PET0301S 3x1 HDMI Switcher with 3D Support - No power supply needed". So I don't recommend that one!
On the positive side, the FTV was able to power the HDMI switch via its HDMI output. That means it should be possible to find a switch that works without needing its own power cord.
So to recap, plugging the FTV into an HDMI input on my Denon 3311ci AV receiver (with no display attached) allows the FTV to output audio on SPDIF optical. Plugging the FTV into this cheap HDMI switch did NOT allow SPDIF audio.

I ordered the in that Ned Scott suggested, which also has an analog port. http://www.dx.com/p/hdv-m612-hdmi-to-hdm...5u-aBaSOz0

This device looks promising for about $15 (except for shipping time...): http://www.amazon.com/CompuLab-fit-Headl...roduct_top

I have my Fire TV plugged into this HDMI switch and it works well; the IR works with my Harmony remote as well.

http://www.amazon.com/gp/product/B00BGBU...UTF8&psc=1