I've gone through the customer support, and they were unable to help. Indeed, I'm returning the Amazon Fire TV box as I have no use for it because of this bug. -- But I do want the Fire developers to know this is an issue, so I'm posting here in hopes it gets to the right developer.
I have a 7 year old 42LG50 television. When the Amazon Fire box turns on, it will display the "Amazon" logo properly, but then the screen goes blank. The unit itself does work on a newer TV set, and I walked through changing the resolution (and remote resolution cycle) to no avail. Also tried multiple HDMI cables, flipped in both directions, and all three of my TV's HDMI ports. The Fire shows the logo, then dies a black death.
I tried a second Fire, thinking the first was defective .. it was not; same issue. Clearly, since the Amazon logo does show up, the HDMI connection is operable for a time, which leaves a software bug on the box as the culprit.
Perhaps this bug can be solved in the 3rd generation of Amazon Fire.