Until Mapbox supports other projections https://github.com/mapbox/mapbox-gl-js/issues/3184 this is going to involve lying about the projection in the metadata source file so that you essentially stick a map with a different projection on the Web Mercator Mapbox map.
If you download coastline from http://www.naturalearthdata.com/downloads/10m-physical-vectors/10m-coastline/ then convert this to GeoJSON in a polar projection (EPSG:3413 which is a projected coordinate system in meters) with:
ogr2ogr -f GeoJSON -t_srs 'EPSG:3413' ne_10m_coastline.epsg3413.geojson ne_10m_coastline.shp
Then fake the coordinate reference definition so it thinks this is EPSG:3857 (web mercator, a projected coordinate system in meters) without actually changing any of the numbers:
ogr2ogr -f GeoJSON -a_srs 'EPSG:3857' ne_10m_coastline.asrs_epsg3857.geojson ne_10m_coastline.epsg3413.geojson
Finally we need to re-project that back to WGS84 since that's what GeoJSON files should be in and it's what Mapbox expects.
ogr2ogr -f GeoJSON -t_srs 'EPSG:4326' ne_10m_coastline.fake_polar.geojson ne_10m_coastline.asrs_epsg3857.geojson
Then in Mapbox https://jsbin.com/junemibegi/edit?html,output
There are a few issues since the source data extends beyond the bounds which our polar projection EPSG:3413 supports, but we could either use a different projection or trim out data to not go so far south.
This works because our polar projection is in meters and centered at 0,0, which is also the case for the projection we used in the -a_srs paramater - web mercator, and the extend of the polar projection used is roughly the same scale but a bit less than web mercator, so our polar map fits into the world.
Best Answer
The current GeoJSON specification https://www.rfc-editor.org/rfc/rfc7946 does not support any other coordinate systems than this one:
Software developed by MapBox is strict with this requirement. Save your data from QGIS as EPSG:4326.
Many software,like QGIS and GDAL continue to support the 2008 GeoJSON specification http://geojson.org/geojson-spec.html that does support different coordinate systems.