If you are permanently resident on Svalbard and are going on holiday to mainland Norway, you must apply for a permit to use your vehicle.
You can be issued with a driving permit for up to three months in any one calendar year. You must apply for a driving permit before you take the vehicle to Norway.
Are you moving permanently?
If you are moving to the mainland, you must obtain customs clearance and pay the appropriate duties for the vehicle.
You will find the rules concerning the temporary use of vehicles registered on Svalbard in the Regulation relating to duty-free import and temporary use of foreign-registered motor vehicles in Norway (Lovdata).
The Directorate of Taxes has prepared remarks concerning the regulation. You will find these remarks in the annual circular concerning non-recurring tax.