I just replaced a water damaged iphone 4S with a similar symptom. Initially the phone wouldn't turn on, but after cleaning everything, replacing the battery, and reassembling, the screen came up with strange blue stripes. The touch sensors still worked and I could unlock, and use with DVI out to a TV. After playing with it for a while, I noticed the screen was coming back to life. It was similar to yours after a few minutes but each time I would lock it, and reopen, the process would repeat (started out worse and "warmed up"). I disassembled, thoroughly cleaned the circuit on the ribbon, and then reassembled to find almost zero change. After replacing the screen ($99) it is now back to a fully working device.