What's Wrong?

Go into the menu of the TV and make sure the "internal speakers" are selected.  Also make sure you have the correct "digital output" selected too.

You need to tell the TV where you want the signal to go.

Both of those suggestions are good ones, but there was no audio problem until I connected the Firestick directly to the HDMI input. The Toslink feed was working fine before, both when the Firestick was connected directly to an HDMI input on the AVR, and when it was connected to a Blackbird HDMI switch that has audio extraction (with its own Toslink output). I get no audio regardless of whether the TVs speakers are selected or not. I will go into the TV menu to double-check the audio output settings as that could be the bugaboo.

It sounds like an HDMI compatibility issue here.  How old is the TV?

2011 Panasonic - never had any HDMI issues before and Fire Stick worked with no problem when I had it connected to a Blackbird HDMI switch for a brief period recently. I have moved a newer Samsung into place and will be hooking that up tomorrow. The Samsung is the reason I got the Blackbird to begin with, because it does not have an optical output, which causes all kind of issues I won't go into here. But my main goal is to get sound without having to use an AVR, whatever that takes. The Samsung is very limited on connection options.