Same problem here as all the above posts. iPhone 6 Plus (10 months old) after 9.0.2 ... I was locked out for period of up to 30mins with the screen unresponsive to touch.
I did an online chat with Apple support. They got me to restore the device from recovery mode (transcript below):
"Once it’s off, hold down on the home button of the device (the one on the front of the device under the screen). While holding that, plug the device in to the computer. Then, KEEP HOLDING the home button until you see first the Apple logo, then the iTunes logo. Then iTunes should prompt you to restore. Just be sure to restore as a new iPhone. Not from back up initially. We want to see if the backup that was attached to the phone, was causing your initial issues. I would recommend having the Phone serviced. We have a few options available. You can take it into an Apple Retail Store or Apple Authorized Service Provider, or we have Mail in service options available as well."
This didn't work. I contacted Apple online chat support again and they got me to repeat the same process. As it still was having issues after restoring the device and setting it up as new, they booked me in for a Genius Bar appointment.
I went to the appointment yesterday. They did diagnostics on the phone and picked up that they're were issues with the display and that there was a slight bend in the device's housing. They replaced my handset and we're good as new.
Hope you all find solutions to your similar issues.