In case of LCD monitors, removing a scratch from the screen is [Most Probably] not possible but for CRT monitors, there is a possible solutions [although not recommended]. And it may have some side effects : )
People do suggest different Simple techniques which they think can help to remove scratches from both LCDs and CRTs but I personally think none of them work in reality.
Anyway, have you heard about Buffing CDs?
The same buffing machine can be used to remove scratches from CRT screens.
NOTE: please never try to remove scratches from LCD via this method.
OK: this is a solution that people use but My Personal suggestion will be to Not use this buffing method for even CRTs. This technique is Not Recommended.