I just checked and no it doesn't. You can probably plug it into an iPod wall charger though, it is just a regular USB cable.
The 360 will go into a hibernation mode when charging controllers, when it is done it will shut itself off. Quit spreading misinformation!
I keep my PS3 remote on my computer desk.. I plug it into a normal mini USB cable and it charges while the PC is on. And, if your PS3 is in standby (red light on), you can hit the middle PS button and it'll turn it on. Very nice feature when wanting to play a game or stream a show from my PC.