If it's an ongoing issue it could be a problem with your proximity sensor. Not sure if that's really the issue or if it's just the way you're holding the phone. You can always visit an Apple or VZW store to have them check it out. Also, try the basics to see if they help if you haven't already. Restart/Reset/Restore.
My wife had this issue last year when she first got her ATT iPhone 4. She was almost ready to throw in the towel. I told her to simply change the position she was holding the phone just a little. Once she did she has never had a problem since. I know my suggestion seems a little odd, but it worked and she refuses to part with her iPhone now.
take joene's suggestion. the screen should black out when it's actually held to your ear.....but if you are moving around, holding it with your shoulder (no hands), or your hair is getting in the way, etc, etc....it messes with the sensor and the screen comes back on, enabling your ear to touch icons.
If you use a case, try removing the case to make phone calls. The case will affect how you hold the phone, which can then lead to the proximity sensor not being activated to turn the screen off while the phone is near your ear.