How to tell if a GeoJSON path intersects with another feature in Leaflet?

You can try the Turf library and a method like intersect: http://turfjs.org/docs/#intersect

Here's the code example from that library:

var poly1 = {
    "type": "Feature",
    "geometry": {
        "type": "Polygon",
        "coordinates": [[
            [-122.801742, 45.48565],
            [-122.801742, 45.60491],
            [-122.584762, 45.60491],
            [-122.584762, 45.48565],
            [-122.801742, 45.48565]
        ]]
    }
}
var poly2 = {
    "type": "Feature",
    "geometry": {
        "type": "Polygon",
        "coordinates": [[
            [-122.520217, 45.535693],
            [-122.64038, 45.553967],
            [-122.720031, 45.526554],
            [-122.669906, 45.507309],
            [-122.723464, 45.446643],
            [-122.532577, 45.408574],
            [-122.487258, 45.477466],
            [-122.520217, 45.535693]
         ]]
     }
}

var intersection = turf.intersect(poly1, poly2);