My iPhone 5 had this problem over the last year or so until last night, when I dropped it from the top bunk of my bunk bed. It was a less than 1.5-metre fall that somehow fixed what I thought was a lost cause of a screen.*
Before, it was just as we described it: it would refuse to respond when being tapped (no matter how hard or how fast I tapped the screen), it would erratically tap, drag, and hold on to the screen by itself. A plus, though, is that it could create some abstract masterpieces by itself in the drawing app. But, other than that, the struggle was indeed a reality. Some of the crimes that my phone and it's notorious screen had committed include failing to launch my camera when I needed it most, not being able to press the green button when I needed to answer important calls, and liking photos for me as I scroll through my feed.
Initially, my only temporary solution for the unresponsive screen was to (again, as many have said), by locking the screen and using the home button to turn the screen on (unlocking with the lock button didn't work). My phone gradually deteriorated and that eventually stopped working as a solution. After months of use, the only solution was to lock the screen and then give it a few solid knocks on its back, just above the Apple logo. This, however, provided relief that sometimes did not last me even to the second digit of my passcode lock.
Now, I'm in complete bliss being able to type this much without having to constantly lock and bang on my phone. There may be a few scuff marks on the corner of my phone, but they are more than worth it.
For those of you desperate enough to try and recreate how my phone was fixed, note that my iPhone is up-to-date with the latest iOS and had no case but had a screen protector on its front and back. It was face-down as I (accidentally) knocked it over the side of the bed. It landed on hardwood flooring and I found it face-down. Good luck.