Power - The first thing you want to do is make sure you are getting the proper voltage to the Dryer. If the dryer is an electric dryer, it needs 220 volts AC to run the heating element. A gas dryer will only require 120 volts AC. Turn the breaker off to the unit and turn it back on. It is possible for the breaker to be kicked and not appear to be. Make sure the cords are completely plugged into the outlet.
Cords - Inspect the cords for burn marks or damage. Check the ends with the prongs that go into the outlet. If they appear discolored, replace the cord. It may become a fire hazard.
Safety Thermostat - The safety thermostat will kill the power to the whole dryer if it detects a problem. It sits on the heater housing. If the temperature of the heater rises above a certain degree, the fuse will trip. It is a non-resettable type. That means once the dryer overheats once, you need to find the cause of the overheating. You may have a failed limiting thermostat or some plugged dryer vents.
I've just learned that this thermostat is in fact resettable. Here is a little trick that you can use to reset this if it has tripped. Simple Dryer Trick Video
Timer - The timer is what send the power to various components on the dryer. If there is a malfunction within the timer, this could cause the machine to not run at all.
Door Switch - If the dryer thinks the door is open, it will not start a cycle. To test the door switch is simple. If the dryer door is closed the switch will show a closed circuit. If the dryer door is open, the switch will also have an open circuit reading. Check for any loose connections on the switch and make sure the lever isn't broke off.
Broken Belt - Some dryers are equipped with a broken belt switch. It detects that when the dryer belt is broke, the dryer will not start. The switch is usually under the motor and is activated by the arm for the idler pulley. If the switch shows an open circuit, the dryer will not start.
Power - It is possible for half the breaker to be kicked. This will allow the motor to run but the heating element will not. This is a very common problem for bad power cords. Blown fuses in the breaker box would be the next thing to go after.
Heating Element - This is the part that creates the heat in an electric dryer. It requires 220 volt AC to heat properly. Heating elements come in all shapes and sizes. They all are basically a coil of wire and glow red.
Igniter - The dryer's igniter is what ignites the gas that has entered through the gas valve. It will glow red, then when it reaches a certain temperature, it will trigger a thermostat that opens up the gas valve and that's when the dryer starts to heat the air. If the igniter has failed, the motor will still run, just your clothes will not get dry.
Flame Sensor - This is the sensor that detects that the igniter is working properly. An easy way to tell if the flame sensor is the problem of not heat is simple. If the igniter comes on then cycles off after about 15 seconds or so, the flame sensor is fine. The problem may be with the coils on the gas valve. If the igniter comes on and does not cycle off at all, then the flame sensor needs replaced.
Gas Coils - These are the pieces that sit on the gas valve that have the wires going to them. These are a very common problem for gas dryers. If the igniter is glowing and cycling on and off, the next thing to replace would be these coils. Some gas dryers have one coil and other models have two. To test the coils you want to remove the wire plugs going to them. Then use your multimeter to test the resistance (ohms) between the terminals on the coils. You want to see a reading of around 1300 ohms give or take 150.
Limit Thermostats - These are the thermostats that keep the dryer's air flow inside the drum around a certain temperature. If one has become stuck open, dryer will think it's warm enough and not allow current to travel to the gas burner parts or the electrical element parts.
Dryer Vents - This problem is most connected to an airflow problem. If the moisture inside the dryer cannot escape, they don't dry. There are two main places the lint can build up overtime. One is the area where your lint filter goes. There is a bend just before the air duct reaches the blower wheel. You can keep the lint filter as clean as you want, but this will still happen over the course of a couple years. To properly clean this area you will need to remove the rear panel of the dryer and remove the cover to the wheel. The blower wheel is what pushes the air and moisture from inside the drum to the external venting system inside your home. If there is a restriction inside the external venting system the clothes will take forever to dry also. It is recommended to have the venting system of your dryer cleaned annually or no longer than 2 years tops.
Belt - Usually when a dryer doesn't spin this is the first thing to check. The dryer's belt simply goes around the drum and the motor pulley. There is an idler pulley and keeps tension on the belt to prevent it from slipping on the motor shaft. If you overload the dryer this may cause too much tension where the motor pulley meets the belt and it will burn a hole and snap the belt. Replacing the belt can be a tricky task. It will require you to remove the entire front panel of the dryer to slip the new belt over the drum. Then you will need properly install the belt around the idler pulley and motor pulley. Another thing that will cause the belts to easily break is a worn out idler pulley.
Motor - Check the motor. You may have to use the wiring diagram under the control panel to check for voltage. If the motor is getting the proper voltage and is not turning the drum, replace the motor.
Idler Pulley - If the bearings in the pulley are worn, this will cause too much tension for the motor to turn the drum. Lint builds up between the shaft and the wheel and takes away the grease. After the pulley runs for a while without the grease it can start to squeak. That's usually when belts start breaking. Replacing the pulley will take care of this.