Option A: Occlusion culling, you will need to find a library for this.
Option B: Use a AABB Plane test with camera Frustum planes and object bounding box, this will tell you if an object is in cameras field of view. (not necessarily visible behind object, as such a operation is impossible, this mostly likely already done to a degree with webgl)
Implementation:
Google it, three js probably supports this
Option C: Use a max object render Limit, prioritized based on distance from camera and size of object. Eg Calculate which objects are visible(Option B), then prioritize the closest and biggest ones and disable the rest.
pseudo-code:
if(object is in frustum ){
var priority = (bounding.max - bounding.min) / distanceToCamera
}
Make sure your shaders are only doing one pass. As that will double the calculation time(roughly depending on situation)
Option D: raycast to eight corners of bounding box if they all fail don't render
the object. This is pretty accurate but by no means perfect.
Option A will be the best for sure, Using Option C is great if you don't care that small objects far away don't get rendered. Option D works well with objects that have a lot of verts, you may want to raycast more points of the object depending on the situation. Option B probably won't be useful for your scenario, but its a part of c, and other optimization methods. Over all there has never been an extremely reliable and optimal way to tell if something is behind something else.