Easter Island is far more remote than either pole. In terms of airports that fly to it (as opposed to other random airports that might be closer) your only two choices are Papeete in Tahiti (2600 miles away) and Santiago in Chile (2300 miles away.) Pretty much all of Canada is closer than that to the North Pole.
As Wikipedia says:
Easter Island is one of the most remote inhabited islands in the world. The nearest inhabited land (around 50 residents in 2013) is Pitcairn Island, 2,075 kilometres (1,289 mi) away; the nearest town with a population over 500 is Rikitea, on the island of Mangareva, 2,606 km (1,619 mi) away; the nearest continental point lies in central Chile, 3,512 kilometres (2,182 mi) away.
There are no airports on Pitcairn. There is one on Mangareva, about 9km from Rikitea.
Now, if the question excludes airports themselves, that is not "how far away can I fly?" but "how far away can I be" then the answer will be on a speck between two very distant airports, such as Easter Island and Mangareva. It's possible you could find something that is 800 miles from each of those two airports, and more than that from any other.
Pitcairn is apparently served by boats from the Mangareva airport, about 1000 miles, so it's looking like a very good candidate for being an airportless place furthest from an airport, whether populated or not.