I've been having the same issue on my 6+ with the touchscreen not responding at all or having phantom touches. My phantom touches were always in a straight vertical line close to the middle of the screen (passing through the "b" key on the keyboard). The problem has been very intermittent -- my phone would go for days without a single issue then have a 24-hour period where it would be completely unusable.
After one particularly bad period, I finally made an appointment to go to the local Apple Store Genius Bar and spoke with a very helpful guy there. He was absolutely aware of the problem and suggested that it could be caused by one of three things:
1) Bad motherboard -- if you are seeing white lines on your screen in addition to the touchscreen issues, then you have a motherboard problem. The only solution in this case is to replace the entire phone. Replacing just the touch screen will NOT solve the problem.
2) Loose or bad touchscreen connection -- According to the guy I spoke with, there are three wires running from the bottom of the touchscreen to the top and then on to the motherboard. One or more of these wires could be loose or broken. He suggested that it might be the middle wire in my device, since the phantom touches always occur in the middle of my screen. If this is the problem, replacing the touchscreen should fix the issue.
3) Bad software -- It is possible that a bad app can interfere with the device enough to cause touchscreen problems. In this case, a factory reset should fix the issue, unless you restore your phone to a backup that includes all the same apps, in which case you'll be right back where you started.
My phone does not have any white lines along the edges of the screen, so he suggested that it was either number 2 or 3 in my case. Since I hadn't yet tried a factory reset, he suggested that I do so and then NOT restore to my backup for a few days, to test things out. If the problem returned on a clean phone, then my issue would be #2 and I'd have to get the screen replaced. If wiping the phone fixed the issue, then it was likely #3 and I could get away without paying for repairs.
I'm now two days into my factory reset and (knock on wood) have not experienced any problems, so I'm keeping my fingers crossed that it was a software problem. I'll circle back and let you all know how it goes.
Thanks to everyone that has commented in this thread.
-S