My iphone 6 developed the crazy touch screen problem about 2 months ago. I searched the forums and realized how common it was. I tried the rubber tape under the metal cover that holds the connections in place but it didnt work for me.
I tried to think of anything I had done that might have caused it and the only thing I could think of was downloading a film from sky store.
My phone is a 16 gig and on checking the memory usage 12.8 was full including 4 used by the sky store app which included the film. I deleted the app which took the memory usage down to 8.8 and hey presto totally cured !!!!!!!
It probably wont cure everyones problem but it is 100% what happened with my phone.