Is there an existing library for converting ArcGIS IFeature types to and from Json/GeoJson?
You should check out GeoJSON.Net by Vish: http://viswaug.wordpress.com/2008/05/11/geojsonnet-a-net-library-to-produce-geojson-output/
It should be pretty easy to extend it to use ArcObjects geometries.
Also, you can take a look at the GeoJSON exporter I did for SharpMap (you'll have to Google that since StackExchange won't let me post a second hyperlink). The geometry objects in SharpMap have a very similar structure to those in ArcObjects so that should be pretty easy to extend as well.
This question with answer from Stack Overflow may work.
I don't see any existing library, but this C# REST interface from ArcDeveloper provides a start. There's also this CC-AO licensed Python toolbox to export a feature class to GeoJSON which should fit the bill, it handles the five major vector types.
The GeoJSON spec is straightforward and shouldn't be too onerous to implement, if the toolbox doesn't meet your needs.
ESRI has this free Python script that converts from feature class to json, geojson or csv. esri2open
A bit of a late addition, but figured it's a useful addition...
I use ArcGIS.PCL which works very well. Allows you to interact with ArcGIS and spit out any of the geometries as GeoJson.