Apparently it's an overheating issue[0]. The 'red line of death' is not a sign that your PS4 is dead, but a sign that it needs to shut down to cool off.
If that is happening immediately on boot, then hopefully the temp sensor or overall temp system is flawed and can be fixed either by a Sony patch or really quickly in their manufacturing process. Hard to believe something like that slipped through the cracks.
Reminds me of Dave Baggett's recent story[1] about how when you jiggled a controller stick during save, cross-talk on the motherboard would end up causing your memory card to become completely erased. I'm no EE but I wonder if there is something really strange at play here ... like perhaps old wiring in your home. I only say this because I can't imagine a really straightforward reason for this to be happening, unless they have a bad batch of temp sensors?
Fortunately iFixit has taken apart the PS4 already, so we can take a look inside the chassis[2]. There's a lot of talk there on the 'revolutionary' cooling system. There's a giant fan/heatsink combination for the CPU, and the exhaust air seems to flow over the PSU to cool it passively. There's another fan beneath the motherboard which looks pretty small.
Could also be a case of left-over tape or manufacturing material, like we saw with some of the Macbooks coming off the assembly line. There was a piece of plastic/tape coviering the exhaust vent that caused the fans to quickly go full speed and the system to get real hot. I can't find a reference to it anywhere though.
In other news, I'm anxiously awaiting my Xbox One pre-order and hope that it doesn't have any of the same issues :)
0. http://manuals.playstation.net/document/en/ps4/basic/partnam...
1. http://www.gamasutra.com/blogs/DaveBaggett/20131031/203788/M...
2. http://www.ifixit.com/Teardown/PlayStation+4+Teardown/19493