I've seen an GSM iPhone 4 do this once before. Customer had it plugged in to charge....Came back a couple hours later and it was off. Wouldn't turn on, iTunes wouldn't recognize, new battery and charge dock did not help....After awhile I stopped trying to get it to work and moved on. Later I was going to put a working board into that frame because of a bent frame I couldn't get straight and the working board wouldn't turn on. I put the board I couldn't get to turn on into the bent frame and it booted right up. It was something in the frame of the phone that wouldn't allow it to turn on. Never did figure out what exactly but if you have another 4s phone sitting around. Try putting the motherboard into that frame and see if it kicks on.
Might be a similar issue to what I had.
Hope this helps!