I'm having the same issues with my new 6+. It functioned fine for about a month then, about 3 weeks ago, the screen became randomly unresponsive.
Over this this period of time I've gone through all the fixes I've found on various messages boards;
soft reset that saves data.
hard reset that completely erases all data and settings.
Restoring the phone from the 6+ back up on iTunes.
Restoring the phone from a previous iPhone 5 backup on iTunes (several times using 2 different backups).
Resetting the phone and starting as a new setup....(several times now).
Sometimes the unresponsive screen can be resolved by pressing the home button. Sometimes it can be resolved by pressing on/off button on the side. As often as not the home button is also unresponsive. Turning the phone off completely can be a challenge....pressing the side button till the slide bar appears might work but if the screen is unresponsive I can't finish the shut off process. At that time, the side button also does nothing. Letting the phone sit for a minute or two results in the slide bar screen reverting to the home screen.
resetting the phone can be equally challenging and can take as many try's as there are key strokes. The last time I rest it took a full 30 minutes to reach the final "reset" and successfully enter the command.
This condition can last from a few minutes to a few days.
The most service I've received from this phone since this problem popped up is about two days, and that has been with minor frozen screen issues resolved by pressing the home button.
My phone has become completely un-reliable and for all practical purposes non functional.
Took it to the Genius Bar in Apple's Reno store yesterday. The tech was very nice and quickly ran the phone through the diagnostics for software and hardware and told me the phone functions perfectly then sent me on my way.
As far as Apple is concerned, at least at this point, is that there is nothing wrong with the phone and it functions as Apple intended.
Of course the phone did not display any unresponsive tendency during their testing but we went to Best buy after leaving Apple to find a cover for my wife and by the time we arrived there the phone was unresponsive and remained that way through several more resets till my frustration finally got the best of me last night.
I deactivated it and re-activated my iPhone 5 with Verizon.
Im at a total loss as to what to do. I'm going to return to Apple this week (i'ts a 120 mile round trip, so I can't just "drop in") and try again but I'm extremely doubtful I'll get any more from them than I've already gotten.
I'm guessing at this point I'll have to try to return it for a refund or at least just give it back to them and refuse to pay Verizon.....then get an attorney and let them fight it out.
Maybe the most distressing thing is that we let them convince us that it's was an easy fix and we bought my wife a new iPhone 6, also on the Edge plan with Verizon.
We switched to Apple about 15 years ago because of the horrible software used by Microsoft. After all those years and two laptops, an iMac desktop, two iPads, and iPhones for both of us beginning with version 4 and upgraded every time a new phone arrived on the market, this is the first time we have had a problem. Apple has always functioned seamlessly and flawlessly....until Steve Jobs passed away. Now it seems the product gets worse every time a new version of the software comes out or a new version of the phone comes out.
We've had great luck with Apple over the years but they have become so unreliable lately that our faith in them has gone away....