I had a similar problem that started almost immediately after updating my iPhone 6 plus to iOS 9.3.2. Touchscreen became intermittently unresponsive. Tried various resets, sometimes they worked immediately, sometimes the screen remained unresponsive for hours no matter what combination of buttons I pressed.
Thinking it was a software problem related to the 9.3.2 upgrade, I signed up to try the beta version of 9.3.3. After installing beta 9.3.3, the screen worked normally - I thought that was the end of the problem. I was wrong - the next morning the touchscreen was unresponsive once again.
Took the phone in to The Mac Store (not an Apple Store) and spoke with the technician. He suggested a "re-seat" of the touchscreen and connectors. Since my phone is no longer under warranty I figured I had nothing to lose. Technician removed the screen, disconnected and then re-snapped the 4 ribbon connectors (that attach the touchscreen to the back plate of the phone) into place (they are located under the top right corner of the touchscreen).
The repair took approx 7 minutes to complete. Technician charged nothing. Phone has worked perfectly since then (48 hours so far). Technician's theory is that the "plus" models are particularly sensitive to slight twisting/flexing of the frame due to the larger size. Eventually the slightest flexing can work a contact loose causing intermittent function of the touchscreen. Seems like a plausible explanation. The repair seems to have worked for my phone.