I do not think it is possible and if it were, it would cost a fortune. Take the train.
The big chains (Hertz has locations at both airports) usually allow such rentals but at very hefty surcharges (think 1000s of NOK not 100s). You'll have to phone them to get a quote they can't be gotten online.
As klimaflyktning said: the trainride is very scenic and definitely worth taking.Edited: 1:27 am, February 10, 2015
There are only 5 hire companies operating out of the airport, there won't be any other major ones in the area - shouldn't take long to give them all a call - https://avinor.no/en/airport/harstadnarvik-airport/to-and-from-the-airport/#!car-rental-962Edited: 3:37 am, February 10, 2015
Harstad-Narvik airport is quite a distance from Narvik. There should be car rentals at Narvik airport and in Narvik town. Just bear in mind that someone has to go to Kiruna and drive the car back to Narvik - that would cost you quite a bit.
Thank you all for your advice
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