Does mist/fog have an LOD effect?

considering it hides objects far away from the player, does it have a LOD effect to increase performance or is it just using up resources to make a simple fog effect?

No, it does not. You still can see the objects so they have to be rendered,

Mist is traditionally used not as direct optimization but to hide the fact that draw distance is limited. In Blender you can set the draw distance on per-camera basis via Clip Start/End and also get/set access in-game via KX_Camera.near and .far.