In that case you narrowed it down successfully - the HDMI port is shorting or the cable is shot. Do you have a replacement cable? Does it short no matter what cable/which end is in? That, and would you mind trying out the composite connector instead of the HDMI, just for testing purposes?
Composite worked fine with the HDMI cable disconnected. Even with the console set to outputting through composite, if I try turn it on with a HDMI cable connected it wont boot.