Same issue here. Phone had a laggy touchscreen for sometime, but never the band at the top and fully unresponsive behavior, that is until I dropped it recently on a tile floor. I found pressing on the upper right side of the top bezel made the problem go away for a short while, as too did removal of the screen protector and case. The problem would always come back, so I decided to take the phone apart, since I am out of warranty.
My first thought was I would open up the screen, remove the plate protecting the screen's ribbon cables, make sure they were all properly attached and then put everything back together. After removing the screen, plate, and pressing the cables down, I tested the screen's responsiveness and everything was working flawlessly. It was not until I made the decision to reassemble the phone that the behavior came back. In fact once reassembled, there was no way to eliminate the band at the top and fully unresponsive screen.
If you are familiar with the internal design of the iPhone there is not a lot of clearance between components. My observations between opening and closing the screen (like a clam), were when closed the screen was unresponsive and while open it was fully operational. My assumption is there must be a short or some type of interference related to the ribbon cables, cover plate, and other contact points on the phone, namely the metal backplate for the screen. I took the ribbon cable cover plate off once more and covered the underside (side sitting directly atop the ribbon cables) with electrical tape. Using an exacto knife, I cut out the plate so the tape only covered the underside of the ribbon cable cover. I punched the holes out for the screws and reassembled the phone.
Guess what. Not only is the screen fully responsive, the aforementioned laggy touchscreen that I have had for well over a year is now fully responsive.
Previously I had tried tuning my iPhone by turning off a series of features, because I thought the lag was due to CPU starvation. When the screen became fully unresponsive, I tried all the remedies I could find, including resetting all settings. The fix described above is the only one that worked. I suspect the reason this is occurring is due to the low clearance in the phone, bending, heat, and something moving a fraction of a millimeter inside the phone. This would explain, why pressure, case removal, and screen protector removal all seem to work for a period of time, because they cause flexion and provide enough clearance to remove the short and/or interference produced by the ribbon cable cover plate.