Fear, Force, Pull, and Push are all considered displacement effects. Effects that negate displacement effects will negate any damage those effects would cause.
It is important to note that while Root means a Champion cannot be moved, it does not negate movement. As the target cannot move, any Push or Pull distance will cause damage as normal. Fear and Force cannot move a Champion suffering root, but do not cause damage, as neither keyword usually causes damage for failing to complete the movement.