The 180th meridian, also known as the antimeridian because it lies directly opposite the (Greenwich) Prime Meridian on the globe, is the line of longitude 180 (positive or negative, the modulus arithmetic works out the same) which runs mostly through the Pacific Ocean from pole to pole.
In theory, it should be a meridian like any other. In OSM, there's two copies at ±180° longitude because the data model doesn't wrap around, so they mark the east-west ends of the world.
Here's the mapnik rendering of the 180th meridian passing through Chukotka.
There's a lake in Chukotka that crosses the meridian at http://openstreetmap.org/browse/way/31254026.
Taveuni and some other Fiji islands are cut in parts by the 180th meridian.
This section should summarize to what extent various tools deal with the 180th meridian gracefully.
Editors treat the 180th meridian as the end of the world, hence have problems editing data that crosses it.
Using some scrolling and zooming, it is possible to edit such data with some degree of success.
- Vespucci : works but currently can not pan or download data extending across the +/-180° meridian - https://github.com/MarcusWolschon/osmeditor4android/issues/389
- Merkaartor : improvement in progress, please check github version. Partially works but may still have some rough edges - https://github.com/openstreetmap/merkaartor/issues/81
- JOSM : older versions had bug(s) which may destroy data when editing objects extending across the +/-180° meridian - https://josm.openstreetmap.de/ticket/2212#comment:10
- Potlach - ??? anyone tested this recently? (22/02/2016 - Potlatch 2 does not render across the 180° meridian and posts an error message "There was a problem loading the map data. Please check your internet connection, or try zooming in.'https://www.openstreetmap.org/api/0.6/map?bbox=%2D180%2E00163229130746%2C%2D16%2E794244779499873%2C%2D179%2E99864055537225%2C%2D16%2E79276186410043")
- Maps.Me : can not pan across +/-°180 (routing not tested)
- OsmAnd : can not pan or route across +/-°180, older versions showed strong map distortion in this area
- neis/heatmap : has problems when 180th meridian is in view (caused by openlayers: http://gis.stackexchange.com/questions/41570/openlayers-vector-layer-markers-disappear-near-dateline)
- openlayers : features disapper when 180th meridian is in view https://github.com/openlayers/openlayers/issues/771
The API map call does not allow for bounding boxes crossing the 180th meridian. There is no agreement if this should be implemented in OSM-API or handled by splitting requests by clients.
All layers on the slippy Map appear to have problems with rendering coastline near 180°, and with ways crossing the 180th meridian. The data browser doesn't handle ways crossing the 180th meridian nicely due to a limitation in OpenLayers.
For GeoJSON data supplied at runtime, Mapbox libraries accept coordinates beyond ±180°, cutting the feature along the antimeridian before display.  However, vector tile data is already cut at ±180° (and along many other meridians).
- Antarctica has special handling of the Antimeridian - https://blog.jochentopf.com/2013-03-11-state-of-the-osm-coastline.html