Kotaku had things taken care of with Sonys help, and it turns out to be a manufacturing miss in some batch of PS4s. Here's a two paragraph quote that really covers the whole deal:
Later in the day, I got a call from Sony. They'd examined our unit and they figured out the problem. . It had been hard to see, though I imagine we would have noticed if we'd examined the unit more closely and not given it back. Nevertheless, we were told that that PS4 had been fixed. The Sony employee testing our unit used a pin to push the small piece of metal back down. They then plugged an HDMI wire into the unit and it worked. They even brought the unit to our office so we could see for ourselves.
The obstructing piece of metal in the formerly-broken PS4 had actually knocked some of the "teeth" out of the HDMI wire—the one bundled with that PS4—that we'd originally plugged into the unit, the Sony folks told us. We checked two other HDMI wires that we'd used during our brief bit of troubleshooting, and sure enough, they were missing the same teeth, too.
Like the title says, watch your HDMI port before plugging in! I hope to get this out to people before they lose hope, or lose cables and hope.