I suppose if the fire marshall decided to investigate, they could potentially trace the fire back to an Xbox. Without a fire department investigation, it would be a lot harder.But it looks like overheating machines are nothing new to Microsoft. I'm tracking the latest on the Xbox 360 at http://www.xboxoverheating.com.
Apparently, the Xbox overheats and crashes, and some guy in Chicago is suing over all of it. I don't know what kind of lawsuit you can file because your Xbox freezes, but it makes good headlines! IMHO, just take it back to the store you got it from and tell them it's broke. Go back to your original Xbox or PS2 and play that until they fix it.
Either that or prop it up on bricks to keep it cool...