Non-vaccinated arrivals from destinations where travel to Iceland is permitted must take a Covid test on arrival, before going into quarantine for five to six days and taking a second test.
New Zealand citizens do not need to apply for a visa to visit the Schengen countries for a maximum of 90 days in any 180-day period. The 90 days can be used either for one long stay or several shorter stays.
Iceland tourist visa is not required for citizens of New Zealand.
Check with your airline in case you have connecting flights overseas as part of your journey to Iceland. It may be the case that countries you pass through en route to your destination may require a separate transit visa. Please refer to visa requirements of the specific country.