Although odometers are very important gauges in your vehicle, for mileage tracking and reference, they are not essential to operating the vehicle. Car engines and necessary parts are not affected if the odometer stops working, although the legalities of driving with a broken odometer vary state by state.
Effects on Other Gauges
When a car's odometer stops working, or slows down, the vehicle owner may have a difficult time knowing how many miles the car is travelling. A broken odometer, however, does not affect other vehicular parts.
Depending on which state the vehicle is registered in, there may be different laws regarding driving with a broken odometer. You are still able to drive the car, but it may not be legal to drive it if the odometer is broken. For example, in the state of California, as long as the car is 10 years or older, it is legal to drive with a broken odometer.
Odometers are significant because when you purchase or sell a vehicle, a large portion of the vehicle's value is based upon the vehicle mileage, which is read off the odometer. Driving without a fully functioning odometer is perfectly safe, but if you plan to sell the vehicle any time soon, have the odometer fixed.