My wife's iPhone 5c had the same issue. Upgraded to iOS8 whatever and soon after started having touchscreen issues. It started out OK initially after the upgrade but then progressively got worse. I did backup and restore, reset to factory then restore, wiped phone clean and setup as new iPhone and it would work for maybe an hour each time. Then the screen would become unusable, not respond to touches/swipes or only do so occasionally. Sometimes white horizontal lines or vertical lines (or both) would appear. You could hit the lock button then home button and get a few more touches in before it would freeze up or act up again.
The fact that it would work fine for a while after restore then slowly degrade made me think the hardware was fine and it's a software issue. I thought: what could affect the screen randomly? Auto brightness sensor? Maybe it would work fine when lighting was even but then start jacking up when lighting changed.
So, I went to settings to turn OFF auto-brightness (this took several screen locks and soft resets to be able to use the screen and get there), I then turned my brightness to the low end (very difficult when screen is not behaving but eventually got it). After doing this, the screen seems to do OK. The randomness of the screen issues after restore must be caused by the light sensor not working properly with the new iOS upgrade.
This is by no means a permanent fix, still going to Apple Store to show them as soon as I increase brightness to full or turn on auto-brightness, the touchscreen issues begin immediately.
Hope this helps others...