Depending on which model Your dryer is it could be the motor, it could also be a safety or hi limit thermostat.
I could get You started looking in the right direction if You can supply Me with the model number from Your dryer.
Looking at Your model the first thing I would check is the door switch, open the door and operate the actuator a few times if it hangs up or does not make a positive clicking sound this is Your most likely part failure.
Other things I could have You check would be the timer and motor but lets see what You find with that door switch first.
Advance timer to an AIR DRY or NO HEAT cycle and start Your dryer, remove lint screen and tell Me is the air pulling into the dryer is it blowing back out at You, or no movement that You can tell?
If there is no air movement or it blows back at You turn dryer off and clean out the vent, and the vent hood outside Your home.
OK if there is no movement then You have a problem with the venting and the dryer is over heating.
Clean does not necessary mean its sufficent the proof is the fact that You did not have any air flow at the lint screen opening, the 2 items I commonly see are the following>
on the outsdie of the home>
and off the back of the dryer if Your pipe looks like this>
either in the foil or a white plastic if You have any of the pictured or similar items You will want to change these to the following>
and for the pipe>
and with improper ventilation can cause problems with the following safety thermostats and fuses>
and depending how long it was ran with improper ventilation this could of also have damaged the heater, motor, and timer which are damaged by the excessive heat and lint. Instead of the dryer being able to pull air from the room across these components to cool them the hot moist dusty air is forced back into the cabinet and into the motor and timer, the lint acts as insulation on the motor futhering the overheating by not allowing the motor to cool properly, and the dust in the timer will cause switch contacts to burn out.
I would have to assume that if the dryer was ran with improper venting for a period of 2-3 years, this dryer has been damaged beyond an economical repair, once You fix one part and get it running, the next will not be far behind.
Hope My answer is informative to Your satisfaction.