My wife's 16GB iPhone 6 - ATT started doing the same thing 2 days ago. Her screen works and suddenly doesn't work. Went back to Best Buy and unfortunately they didn't have inventory, but I'm on the list for a brand new one since it's still in the first 2week period.
Interesting tho that i called apple did all the troubleshooting stuff they usually ask to perform and the guy said that there is two option:
1, send the phone in for service (my wife would stuck without a phone) and they would either fix it or send me back a refurbished one. Not a brand new one!!
2, Express service for $30 they send me a phone that i could keep until they fix or send a refurbish one.
So I pay full price for the phone and it's still under warranty and it's their fault pretty much and they want me to send a refurbished phone not a brand new one. That's a shame on apple! Unfortunately the closest apple store is 3hours away from me. So i'm hoping that Best Buy will call soon.