I have 100% the exact same issue.
Mine started the same way too. Screen became unresponsive but then worked again a while later so I thought nothing of it. It happened again and then again. Eventually the phone would go an entire day being entirely unresponsive to touch.
I contacted apple support and they recommended...
- Restarting the phone (sometimes fixes it, sometimes doesn't)
- Backing up and restoring the phone (seems to resolve things for a little while whilst it restores but soon bad again after a day or so)
- Wiping the phone entirely and setting up as a new phone (worked for a day or two then same occurred) - This effectively eliminated a software issue as the cause.
Eventually I took the phone to the Apple Store (I waited 3 hours for an appointment). They ran some tests and stated that it was a 'touchscreen accuracy issue'. They asked if I had a backup then took the phone in for repair. I traveled the same 40 miles it takes me to get back and pick the phone up. They had replaced the screen and said I was back up and running.
3 days later, same issue so I booked an appointment and waited another week to see someone only to be told that because the screen had already been replaced that they should just replace the phone then and there which they did. Brand new phone.
So I restored the phone with my iCloud backup in store then and there and all was OK.. now about a week later AND after updating to 8.4.1 every now and then I get grey/white, flickering at the top of the screen encroaching in to as much as a 5th of the screen itself. The touchscreen becomes mostly unresponsive, occasionally opening apps at random like I've touched it in different places to where I've really touched it.
This new trick can last anywhere between 5 seconds and 5 minutes and seems to (maybe) be linked to the accelerometer because sometimes when I go to show someone what it's doing it immediately stops.
I'm fairly convinced this is a software issue and NOT a hardware issue.
I too suspect that an app I have on the phone is cause this, but no idea which one. Maybe we need to get together and cross reference our apps to find a lowest common denominator?
The ones I have that I know most other people don't are:
Waze
Pebble
Pebble Time
Alien Blue
Imgupr
AroundMe
RingGo
My VINCI Park
Bebot
White Noise
Sleep Talk
iZIP
TV Remote 2 (Panasonic)
Smart View (Samsung)
Smart View 2 (Samsung)
TV Guide
LogMeIn
LogMeIn Ignition
KOTOR
Super Smash Bros. WIKI
OWA
Office Mobile
Lync 2013