I've flown directly over the north pole on a route from ATL to PEK (beijing).
Here's a pic of the seatback flight map from a United 747, where I spent the entire flight with my face pressed to the window.
Amazing scenery from Hudson Bay north through Baffin Island and on.
It was July of 2008.
When we crossed the pole (as indicated in the flight map) the icon of the airplane began to flip back and forth, not knowing exactly which way it was pointing.
I thought that was really cool and shot video of it trying to establish it's direction over the course of about 3 mins.
Once we get a little bit of distance from magnetic north, the icon settled down and pointed toward Beijing.
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In order to fly over the north pole (or more broadly just the Arctic), you'd have to travel to a city that is approximately across the globe as then the shortest distance would go over the north pole.
You can try out which cities satisfy this criterion with the distance tool of Google maps.
One possible route is for instance from the Middle East to the American West Coast, for instance this flight: Dubai to San Francisco .
A little more information about this can be found on the Wikipedia article Polar route and it seems those routes are fairly common.
However I think being based in Europe, there are not many flights over the Arctic.
Such a route from Europe would lead to the Eastern tip of Russia directly or to some of the Pacific islands, where there's probably only very few direct flights to.
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Building on @user34936's answer, it seems Condor's direct flight between Frankfurt-am-Main and Anchorage could fit the bill.
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Obviously the exact 'north pol'e could be as big or small as you wanted - is it a pinpoint spot that you must fly over, or a larger area, for example.
Wikipedia now has a list of polar flights today , as defined officially as polar flights.
However, it's also worth noting that the definition of 'polar flights' has changed: The American Federal Aviation Administration now defines the North Polar area of operations as the area north of 78 deg north latitude, which is north of Alaska and most of Siberia.
The term "polar route" was originally more general, being applied to great circle routes between Europe and the west coast of North America in the 1950s.
There appear to be several routes that fit this criteria: Polar routes are now common on airlines connecting Asian cities (Bangkok, Beijing, Dubai, Guangzhou, Hong Kong, New Delhi, Seoul, Shanghai, Singapore, Taipei and Tokyo) to North American cities (New York, Chicago, Detroit, Houston, Los Angeles, San Francisco, Seattle, Toronto, Vancouver and Washington, D.C.).
Emirates flies nonstop from Dubai to the US West Coast (San Francisco, Seattle and Los Angeles), coming within a few degrees of latitude of the North Pole.
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How close to the north pole do you want to be?
Since it is just a where the Earth's axis of rotation meets its surface , you'd have to be pretty lucky to fly directly over it :)
Besides, transcontinental flights usually try to take advantage of the jet stream , so I think you'd be better off renting your own plane and crew.
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I always like flying to/from north America, since the route usually goes over Greenland. I do like the idea that I am only 10 km away from this  inaccessible  land. I would love to be able to say that I have been that close to the north pole. Unfortunately, so far my journeys remained to the more southern parts. Do airlines fly directly over the north pole and if so what route should I take to have the best chance?
Do airlines fly directly over the North Pole?
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In order to fly over the north pole (or more broadly just the Arctic), you'd have to travel to a city that is approximately across the globe as then the shortest distance would go over the north pole. These polar routes are listed on Wikipedia and are most common on flights connecting Asian cities to North American cities, as well as from the Middle East to the American West Coast. Flight that pass over the North Pole are usually those between Asian and North American cities, e.g. Emirates flight between Dubai and San Francisco. To find out which flight to get on, you can now check Wikipedia that has a list of polar flights.
The main routes that fly over the North Pole are those from Asian cities to North America. There are several airlines that fly these routes including Emirates. To fly over the North Pole you need to fly to a place on the other side of the world from where you are. Wikipedia now lists polar flights.