Will a bad temperature sensor make your car run hot?

It can if your sensor is hooked up to your fan. What happens is that the signal from the temperature sender turns your fan on, or doesn't if it's not working. If your fan isn't turning on when it should your car will run hot. To check your temperature sender disconnect the plug from the sender and test it using an ohm meter. When the engine is cold you should have infinite resistance. Then run the engine until it gets warm. Shut off the motor. Disconnect the plug from your temperature sender and check the resistance again. It should no longer be infinite ( see your repair manual for the correct reading).