Don't have my 3 yet, so not sure what the screen is made of, but assume it would be similar to cell phones or tablets, so maybe something like type of glass. Might be hard to totally remove it, but might at least make it not as noticeable. Might just require a bit more elbow grease with the toothpaste. I'm sure different brands can be slightly more abrasive, so worth doing some research.
Good luck! Let us know if you have any success.