Speed Queen Dryer Model ADE3SRGS173TW01 - when I open the door to remove dried clothes, unless I turn the timer dial to Stop, it clicks loudly, like a bad clock. Does this need to be repaired?

It is likely to be one of the thermal fuses. You can check them with a multimeter for continuity.

https://www.google.com/search?q=Speed+Queen+ADE3SRGS173TW01+parts

Speed Queen wiring. Nothing specific to the model.

https://www.google.com/search?q=Speed+Queen+wiring+diagram

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The load is dry, yet the dryer keeps running and running. Before that unnecessary heat damages your clothing and drives up your utility bill, you may need to replace the timer or the electronic control board that isn't ending the cycle. If your dryer uses a mechanical timer and doesn't stop when the dial reaches the off position, then a failed contact in the timer is likely causing the dryer to continue running and you'll need to replace the timer. If your dryer uses an electronic control, a stuck motor relay on the electronic control board can cause the dryer to run constantly with the dryer door closed. If the dryer stops only when you open the door, you'll likely need to replace the electronic control board.
If the dryer won't turn off when you open the dryer door, the dryer door switch likely failed. Replace the door switch if it doesn't accurately detect the door being open or closed.
These tips can help you figure out what's wrong when your Kenmore or Kenmore Elite dryer spins when off. You can also use this repair help to fix your Whirlpool, Frigidaire, Samsung or LG dryer that runs when the control is in the off position.

IT is the timer. If electronic needs to replace the board. If dial type, needs the timer

if you have it on time dry and it doesn't count down the timer is bad but it should stop when you open the door unless you have a broken door switch too,just open the door and push in on the switch with your finger,it should make a clicking sound,if not the door switch is also broke,if you have the timer on auto or more or less then it won't move without heat.on auto the heat comes on when the dryer reaches temp. the heat shuts off and the timer will advance a little,then the heat starts again and the timer stops,it will do this until the clothes are dry by using a sensor instead of time.

Clothes not drying, but dryer runs & heats
1. Defective operating thermostat---test thermostat,, replace if defective
2. Fan or it's pulley loose on shaft--tighten set screw
3. Incorrect heat or timer setting--reset timer
4. Clothes too wet when placed in dryer--wring clothes before putting them in dryer
5. clogged lint screen or duct ( little air motion)---clean

try cleaning the moisture sensor bars just inside the door and below the door,with isopropyl alcohol of course after unplugging the unit for fire safety,these 2 chrome colored bars detect the moisture in the clothes and when they are wet they slow or even completely stop the timer,as they dry they let the timer again move,if they get a film on them they will not operate the timer correctly,they work on capacitance

How old is it?

If your dryer seems to run forever, it could be because of a clogged vent or internal ductwork. Your dryer may have an automatic cycle that turns off the dryer when the clothes are dry. It does this with a special thermostat or moisture-sensing system.

Normally, this is what happens during an automatic cycle:

• The thermostat tells the dryer to heat until the interior of the dryer reaches a pre-set temperature--say 135 degrees.
  
  
• When the dryer reaches the pre-set temperature, the thermostat tells the timer to begin advancing. (If there's a moisture sensor, the timer advances only if the moisture content of the clothing is low enough.)
  
  
• The timer advances until the interior cools, then the thermostat tells the timer to stop advancing, and tells the dryer to start heating again.
  
  

This cycle continues until the clothes are dry. But…if the vent is clogged, the dryer may never reach the proper operating temperature, so it doesn't send the signal to the timer and the dryer continues to run indefinitely, even if the clothes are completely dry. To fix the problem, clean the vent and/or internal ductwork.

There should be a timer motor on the timer. That is most likely burnt out and will cause the timer to not shut down. Just replace the timer and that will solve the problem

YES MY DRYING IS HEATING UP. THE TIMER WILL NOT SHUT OFF AND STOP THE MACHINE FOR DRYING IF USE ANY TIME.