I had an Iphone 6 plus (model LL/A) with the same problem as you are.
I don't know exactly the mistake was come from firmware or hardware. However, as my experience now, i think it is caused by firmware.
When this problem happen to me, the iphone was at IOS 9.3.3, i tried to shake, lock and open screen, restart, press on some edges of the iphone even bang it softly then the screen respond correctly some times but this problem happen more frequent and i could't used the screen although i tried all above actions.
Then, I called to the Customer care of Apple, they asked me to bring my iphone to the care center for checking and fix. Because my iphone was jailbroken, i had to restore its firmware. I restored to IOS 9.3.5 due to IOS 9.3.3 was signature closed.
After the restoration, my Iphone 6 Plus worked very well in 1 or 2 days before the problem came again. This time, I tried my luck with IOS 10 (14A5346a). And boom, my Iphone have been well since last Wednesday 7th Sept.
I would like to share this with you and all technicians of ifixit.com with the hope for a completely cure to our Iphone. Apple has been revoked us, we have to help ourselves.
Hope this can help many people in the world.
(Please don't charge for fixing this problem by my shares)
This is my original post: iPhone screen flicker and touch is not working