First, try to reset the phone by holding the sleep and home button for about 10sec, until the Appel logo comes back again. You will not lose data by resetting.
If this does not help, and to make sure that this is not software related, set it back to factory settings, without using any backup data afterwards. Set up the rest of the personal settings manually and test the screen. If it still does not work, this is a hardware issue and the phone has to be serviced by Apple or an Authorized Apple Service Provider:
Use iTunes to restore your iOS device to factory settings - Apple Support
Find an Apple Authorized Service Provider
iPhone - Contact Support - Apple Support