No it doesn't.
But it does hide it whenever your cursor is on top of the game as to not interfere with the gameplay, as well as try to keep the cursor in the center of the screen (that's how first-person cameras work). Pausing the game will "unstick" the mouse, so you can move it outside the window.
Due to an oversight the intro menu also tries to keep the cursor in the center even though it is not needed at that point.