Insurance points are placed on you by an insurance company. Thus it will depend upon your insurance carrier's rating system and insurance points guidelines, as well as state laws or regulations, how long insurance points will stay with you.
In many cases the impact of older insurance points lessens over time. If your company adds insurance points for various traffic violations and accident claims, as time goes by and you receive no new violations and file no more claims it will appear that you are a safer driver and these insurance points could be reduced.
To find out about your insurance company's rating system ask your agent. For information about your state's insurance laws contact your state insurance regulator.
DMV points are what are placed on your driving record and will remain there for as long as your state laws require the Department of Motor Vehicles to keep them there. In some states this could be 3, 5 or 7 years while in other states points are not reduce or dropped off at all. To find out how long these points stay on your driver's record contact your local DMV.