renew

Doing an actual Renew directly from Norton you will not lose those remaining days.

But if you are purchasing a new subscription key directly from Norton, you will lose those days. Norton starts your subscription period as soon as payment has been processed. There is something you can do about this. After the new key purchase, you can contact Norton Support and ask them to add the time from the new key to your existing subscription. 

If you are purchasing a new product key from a reputable retailer, you can hold that key until one or two days before expiry of your current subscription, and enter the new key then. Or you could contact Support and ask them to add the time from your new product key to your existing subscription.