A Design Defect Is Breaking iPhone 6 Pluses
I had this same issue with my iPhone 6 Plus screen not responding to touch as this article explains. Took it into Apple in Perth City (Western Australia) and they replaced the iPhone on the spot with a brand new refurbished. This article leads with a story about apple denying it to a customer, but I had my issue resolved within 25 minutes including running a last minute icloud backup. Apples legendary support is real.