Why isn't my touch screen responding?

I've had a refurbished iPhone 6 for a year and a half. It works fine for the most part, but I'm having some issues with the touch screen responding. My screen just stops responding out of the blue, no matter what I'm doing on it. I have tried restarting it (with the home and power buttons).

What does seem to work is literally pushing the screen down onto the back of the phone. It usually works fine for a little while after that before I have to do that again. This happens about 5 times per day. I'm guessing it has something to do with the screen itself and not the touch response.

Has anyone else had this problem? Is there anything I should try before just getting a new phone?