Screen is unresponsive and only sometimes turns on.

Touch Disease is much less prevalent on the iPhone 6 versus the iPhone 6Plus.

Unfortunately, the answer to your question is that it could be a defective screen or Touch Disease. You really don't tell us enough to make an educated guess. Has your phone been dropped recently or has the housing been bent in the past? Does your phone exhibit grey bars at the top of the screen? This are things that could cause Touch Disease.

The more likely scenario and the easiest one to eliminate is that you have a defective screen. Ideally, you need to try a known-good screen to see if it works better or not. You could find one/buy one yourself or go to a local shop so that they can try one. It might also be related with the connectors. If a known-good replacement screen makes no difference, then you likely have a logic board issue. That will require a micro-soldering repair and is not DIY.

You can check out my profile for a link to my blog post on Touch Disease if you want to learn more (or you can search the web, there are plenty of good resources).