Iceland to North Pole vs. New Zealand to South Pole: The Clear Winner

Iceland to the North Pole: ~2,700 km (1,700 mi).
New Zealand to the South Pole: ~4,300 km (2,700 mi).

Iceland wins by 1,600 km. Reykjavik sits at 64°N—only 26° from the pole. Invercargill (NZ's southernmost city) is at 46°S—44° away. The Arctic Ocean's geography lets land approach the pole; Antarctica's continent pushes the coast north.

Bonus: you can fly from Reykjavik to Longyearbyen (78°N) in 3 hrs. From Christchurch to McMurdo (78°S) takes a military C-17 or icebreaker—weeks, not hours.