According to Newton's third law of motion, which states that "every action has an equal and opposite action, provided the action and reaction forces act on two different objects", yes.
Although, it is not exactly visible to prove such a fact, a simple example to prove this fact: For instance, when a toy car hits a wall, you see the toy car moving backwards. This is because of the reaction force exerted by the wall and the action force is exerted by the toy car.
Another example: In aviation, jet propulsion totally involves Newton's third law; a jet engine propels a burst of both combusted and fresh (not combusted) air to provide thrust (force in perpendicular direction to anything, in this case an airplane). Here, when compressed air (that is a whole another story of a variety of compressors, etc. inside the jet engine) is propelled out, it pushes air particles behind. These air particles then push the plane in the forward direction (Newton's third law of action and reaction forces) and this makes the plane move forward, hereby "propelling" it. Here, air particles provide the reaction force while air released from the jet engine provides the action force.
So basically, non living things do exert force.
