I looks like your board's voltage regulator got damaged therefore when you plug it there's short circuit which causes the heat. I'd suggest you to submit an email to Intel® Support.
I don't think it will help me...
Warranty is only one year and I have it more than one year. And even if it was still under warranty, I got it from friend from other country so sending it would be problematic.
I myself don't have the things I would need to repair it myself, and sending it for repair + repair cost would liklely be close to (or exceed) the cost of new galileo gen 2 board.
I read this: Galileo Gen 2 Power IC Damaged(Power on Galileo Gen2 by 5V external supply), the same part was getting so hot so I tried to power it directly from 5V source. The "ON" led still turned on for less than a second and then nothing. I measured the current to see if there is some short circuit - current was something around 0.1-0.2A. So it doesn't seem to be that.