I had the same problem, and have now made multiple weekend trips to the genius bar.
Symptoms: started with a non-responsive or inaccurate touch screen. Then, would get grey flickering... As the problem progresses, the phone randomly presses buttons if it is unlocked.
Here was my experience:
Visit #1 - they replaced the screen.
Within 24 hours, the problem returned.
Visit #2 - they replaced the phone....
Within 48 hours, the problem returned.
I've contacted chat support, and their next question was: could the problem have been created on the new phone after having restored from an iCloud backup.
Their best advice and next step was to plug the phone into iTunes, and do a full reset.
Incidentally, this has not solved the problem... (It's also a step that I can't recommend -- because it's pretty difficult to set up a phone as new....if you can't reliably use the touch screen.
So, I'm going on a month with a useless phone, and the best that Apple support has been able to do for me is continue booking ANOTHER genius bar appointment. Where they'll replace the phone....and I'm guessing the problem will come back.