Do you have the other AV cable (the one that has Component and Optical outputs) plugged in as well? This guy was having a similar problem to you, and it turned out that the Xbox was switching output modes:
From what my understanding is, the Xbox automatically detects when you switch to and from HDMI . And if you have the AV cables hooked up also it switches to the next connected mode which is: Component. The problem is the Xbox requires a restart when it switches modes.
If it's not that, it might be a feature of your TV called CEC that you can probably disable:
Consumer Electronics Control (CEC) is a feature designed to allow the user to command and control two or more CEC-enabled boxes, that are connected through HDMI, by using only one of their remote controls (for example by controlling a television set, set-top box, and DVD player using only the remote control of the TV).
Each TV vendor calls this something different in their options, but the Wikipedia article I linked contains some popular brands and what they call their version of this feature:
Trade names for CEC are Anynet+ (Samsung); Aquos Link (Sharp); BRAVIA Link and BRAVIA Sync (Sony); HDMI-CEC (Hitachi); E-link (AOC); Kuro Link (Pioneer); CE-Link and Regza Link (Toshiba); RIHD (Remote Interactive over HDMI) (Onkyo); RuncoLink (Runco International); SimpLink (LG); HDAVI Control, EZ-Sync, VIERA Link (Panasonic); EasyLink (Philips); and NetCommand for HDMI (Mitsubishi).
Just in case you can't figure out why it's turning off, there's an option in the Xbox 360 preferences where you can specify if you want the console to stay on and continue a download even if you power it off. In situations where you just want it on so that you can grab a big item off of the Microsoft servers, you can just set up the preferences properly, and then it will finish downloading regardless of whether or not you turn it off after you start the download.
How can I turn off my TV without turning off my PS3?