Q: iPhone 6 screen unresponsive

Did you try to set it back to factory settings, without using the latest backup?. If the screen works again, restore from the backup.

Use iTunes to restore your iOS device to factory settings - Apple Support

If factory resetting does not help, get the phone checked by Apple, this might be hardware related.

Start a service request

Find an Apple Authorized Service Provider

Make an Apple Store appointment

I have this issue too, big time. My iPhone 6 touchscreen will not respond 80-85% of the time now.

They call it IPhone 6 "touch disease", due to fragile design of the touch IC chips.  Let us know if you decide to take it to Apple. The company needs to address this issue rather than treating otherwise loyal iPhone 6 owners like scum.

You can also choose to visit an Apple Authorized Service Provider next to your place, they can also check your phone and provide service, if this is more suitable.