No you don't have to render 240 times per second if your game doesn't require it.
You should always limit or offer the possibility of limiting of both the rendering and logic of your game.
Here is the POC: https://jsbin.com/cuwupaz/edit?html,js,output
What values do you see? Move the window to another monitor, does it stay the same?
Go to lines 33 and 38 and uncomment them to limit your fps.
* do not render excessively if your game doesn't needed it
* decouple your game update logic from the rendering
* do not hook the update logic to the monitor refresh rate!