I had a very similar issue. One morning, my screen on my iphone 6 just became unresponsive. My screen was not physically damaged at all, no scratches, breaks, cracks, anything. 1/10 times I tried to use the touchscreen, it would work, but most of the time it left me unable to enter passcode (after restarting so fingerprint wasn't enough), answer calls, or do anything. I tried restarting (which I regret because then I was required to enter my passcode and couldn't do that because the screen was still unresponsive). I also tried restoring my phone to factory settings via itunes. I tried to "slightly bend" my phone as some people had stated in previous posts and convinced myself that it worked once (because I could enter my passcode before the screen became unresponsive again). But overall, nothing was leading to a long term fix.
I went to a phone repair booth in the mall (yes, this is real) assuming I needed a new screen (~$120), even though mine looked perfect physically. The man working at the booth spent quite a bit of time with my phone to get a proper diagnosis. He opened up my phone and switched back and forth between my screen and a brand new screen. When the screen (either new or old) was connected with the phone still open (not screwed closed), the screen was always responsive. However, when he put the phone completely together again, the screen became unresponsive.
He ended up finding out that it was a simple grounding issue. There's a metal plate that covers the connections from the screen to the "board" within the phone. This metal plate is held in place with 5 screws - 4 at each corner and one in the middle. The kind worker spent time adding one screw at a time, then testing the screen's response. He found that when adding 2/5 screws, the screen became unresponsive. Without these 2 screws, the screen worked perfectly fine. So, he simply removed these two screws that were causing the screen to become unresponsive, closed up my phone, and it works perfectly! This whole ordeal only cost me $20 diagnostic fee. This fix was 1 week ago now and I haven't had any issues so I really believe it was a grounding issue.