I am having the same problem. First time I went to the Apple store (my appointment was 3.5 hours behind schedule) they replaced my phone with a new one. It worked fine for about two days then it started happening again. I could not believe it. I went to the store a second time, just 4 days after they had given me the new phone, and this time they said it was a hardware issue so they replaced my screen (this was a full day repair).
It worked well for about a week and then it started showing the symptoms once again: screen starts lagging, then it doesn't respond to touch but there's typing and moving through pages without me even touching the screen, and ultimately a completely unresponsive screen.
I am going to the Apple store later today to see what they say and I do not think this is a hardware issue. Another thing that worries me is that my 6 month warranty is almost up, so what happens then when my iPhone 6 inevitably goes unresponsive and I am no longer under warranty? Hopefully Apple is working on a fix for this because I can't keep taking days off from work to go the the Apple store, plus I shouldn't have to go without a phone for hours at a time because of a problem they don't seem to care about.