If low power didn't kill electronics we wouldn't have to worry about brown outs, but we know that brownouts do kill electronics.
In terms of PC components, I don't think it is the actual GPU or CPU that gets damaged, but rather the VRMs. They are designed to take 12v +/- 10% in and output whatever voltage the GPU/CPU needs. But if the voltage going into the VRM drops the amps go up, and the VRM has to work harder to supply the component with the power it requires. This could damage the VRM.
But the more likely reason as to why GPU manufacturers put minimum power requirements on their cards is so the power supply isn't overloaded. If the power supply is overloaded, then it dies, and when a PSU dies it could take components in the system with it. Of course an overloaded PSU will also likely have pretty big voltage drops, so if you start feeding the VRM 11v when it is expecting 12v, it could damage the VRM.
Rig1: System Specs.
Rig2: A8-5600K@4.4GHz / AsRock FM2A75 Pro4 / 8GB Corsair DDR3-1600 9-9-9-24 / HD7560D / Samsung DVD-Burner / 1.5TB WD Green + 3x3TB WD RED in RAID5
Rig3: Athlon X2 4200+ / M4A79 Deluxe / 4GB G.Skill Pi DDR2-800 4-4-4-12 / GT430 / Sony DVD-Burner / 500GB WD
Rig4: Phenom II x6 1605T @ 3.6GHz / Asus M5A99X Evo / 8GB PNY DDR3-1600 9-9-9 / GTX470 & GTX470 / Samsung DVD-Burner / 1.5TB Seagate