In short, it shuts itself down. One NeoGAF member discovered the new error screen and immortalized an inevitable meme by snagging this quick shot.
At the time of the error, the Xbox 360 lit up with the new Red Dot of Death (the replacement to the old Red Ring of Death) which, apparently, doesn't necessarily signal death but impending death. When the new Xbox 360 overheats, the center dot will glow red and the system will shut down.
Ironically, wrapping older gen Xbox 360s in towels was a well-known means to fix certain problems with the systems (most probably because it melted and mended broken connections). But the new Xbox 360 will evidently shut off when it detects temperatures growing too high, meaning if hot boxing might still be a means to repair the new Xbox 360, the option is no longer available to users. [NeoGAF via Engadget via Kotaku]