It's possible, but it would have to be a low-power hard drive and it certainly would do very little to help. Most of the heat produced by an Xbox 360 (And other desktops and laptops) is coming from the CPU and the graphics.
Better to just get a fan and move the Xbox to somewhere where heat dissipation would not be an issue such as an open space and having it sit on linoleum, hardwood, tiles or on a desk rather than having it sit on carpet.