I was having the exact same issue on my iPhone 6 Plus 128GB since iOS9. In fact, after doing the following per APPLE Support, I still had the issue.
1. Backed up phone
2. Reset to factory defaults (erase all Content and Settings)
3. Restore Phone from Backup
4. Test the phone and with in a few minutes my touch screen would not respond. Lock & Unlock issue gone for the time being.
Again, per APPLE support, did this
1. Reset to Factory defaults (erase all Content and Settings)
2. Setup as a new phone.
3. Before I could even get by the "HELLO" screen, the touch screen would not respond. I had to LOCK and then UNLOCK the phone to get by the very first step.
Apple support said I had faulty hardware, which, I thought, ok, never had issue with the phone until iOS9. Never dropped the phone, never got it wet, but what ever. So, Apple wanted me to take it to an apple store to have the screen replaced for $125/$150 (out of warranty). Fortunately, I have the AT&T protection plan and I have not used it yet since I have had it (over a year). So, I call them and they sent me a BRAND NEW iPhone 6s Plus 128GB as the replacement (for $99). So, I got the phone last night. I set up as a new phone, upgraded to iOS 9.1 and then restored from my backup. This morning, I am starting to notice the same issue on the iPhone 6 Plus phone. However, the touch screen seems to only temporarily become unresponsive, but I can tell there is still something wrong.
So, I find it hard to believe that both my original iPhone 6 and the brand new iPhone 6s have hardware issues. Also, when talking to AT&T support, they seems to be aware of this issue with iOS 9.x, but it seem APPLE thinks it's an hardware issue. Hopefully, with the next iOS update they will fix this and they are not coming out and saying yes, there is a problem.