I recently had repaired an iPhone 6 plus with a non responsive touch screen. I tried replacing the display and the new display did not respond to touch either. I then tried restoring the phone with no success. I did a little research and found the solution to the problem. For me it was definitely hardware related. This is the article that helped me out. https://ifixit.org/blog/8309/iphone-6-plus-gray-flicker-touch-death/
The customer said at some point they did see the grey bars on the display. When they brought it to me the gray bars were not present. I noticed that if I bent the back casing in just the right area the touch screen would respond to touch. So after reading the article I mentioned, I started thinking it had to be cracked solder joints on the touch ICs. I then decided to reflow both touch ICs and even the digitizer connection on the motherboard with my hot air rework station. This did get the touch screen working properly again. I did not have all the tools at the time to remove, reball, and replace the touch IC. I do suggest having the IC replaced or at least reballed with new solder. Reflowing the IC may fix it for you but in time the solder joints may become week and crack again or they may not if your lucky. So you may want find a repair shop that can actually replace ICs on the motherboard if you do not have the tools or knowledge to do it yourself. You can try reflowing the ICs too, just keep in mind it will likely be more of a temporary fix compared to removing and reballing the ICs with new solder.