As far as I remember, the 747-400 had an operational limitation which prohibited flight at latitudes exceeding 89 degrees. In other words, one had to miss the pole by at least 60nm. My guess is that this was to avoid the problems experienced by the unfortunate CAF Aurora crew.
Cessna Citation and CJ models with which I am familiar have a “keyhole” shape around the North Pole and Hudson Bay within which flight is prohibited. As they have AHRS but not INS/IRS, they rely on magnetic sensing to feed the compasses and I think that the large magnetic dip values in this “keyhole” reduce the magnitude of the horizontal component below a value that is required by the flux sensors.
I did once accept a direct routing from YFB towards YYQ which just nibbled inside the edge of this keyhole, but the wings didn’t fall off.
There is a “DG mode” which removes the magnetic sensing input and then requires the pilot to manually align the displays with the wet compass, just like a Cessna 150.