The filter will not necessarily shut the furnace off, but it will cause the furnace to work harder and use more energy, which can lead to premature failure of the furnace.
It is important to change your furnace filter on a regular basis, especially during the colder months when your home is using the furnace more often. A dirty filter will restrict air flow and reduce the efficiency of your heating system. It can also cause excessive wear and tear on the equipment, leading to higher repair costs down the road.