The USB Type-B port on an UNO went nuclear hot on me yesterday after plugging and unplugging it as an Arduino ISP for my ATTIny burner board. I've done this hundreds of times.
My question is, if this has happened to anyone, did yuo tear it apart or just call it a loss and trash it?
That generally implies either a short or an external power supply fighting with the supply of the USB port (particularly if the grounds of the two supplies were both not-isolated, and at different potentials). Could either of those have been the case? Regarding possible short - one situation that has happened to me is that there was a little blob of solder on the desk, and a board got put down such that it shorted power and ground.
Does the board work with the burner shield removed? (make sure the USB port still works!) If not, I'd say it's aced and pitch it. If it does seem to work, feel the SOIC-8 package (opamp) - it often fails when power rails are abused, and when it does, it fails to a near short, eating a few hundred mA of current and getting hot. If it's hot, it needs to be replaced.