I use the ST_Simplify() function in PostGIS, and am moving to a new geostack with mongodb. Does it have a comparable function?
No - there are three (well looks like two really) GeoSpatial commands in MongoDB. One is nearest and the other is search. I guess simplification is left as an exercise for the reader.
You could simplify your data before saving them into mongodb. I use the node.js mapshaper library to achieve this.
The main challenge (in my view) with simplified geojson data is that mongodb does not support more than one index pr. collection.