My Asko T721 Electric Dryer is having drying troubles

Here's the answer:
The dryer has a thermo protection "fuse" that pops-out when the lint filter is not cleaned and air will not flow properly. When the airflow is restricted the dryer gets too hot and the "fuse" pops. Then it will not run unless you stand there and hold the start button in. As soon as you stop pressing the button the dyer stops running.
Clean the filter ESPECIALLY the "internal" filter. This one is behind the lower skirt of the machines face. There are two Torex head screws (T20) to be removed and the skirt comes right off.
Inside you see two red handles that you turn to open the filter door. Open the door and pull out the filter. It is a long metal finned box. Take it to your bathtub and run a ton of water through both ends to remove the lint inside. You'll be surprised at how much.
O K now let's "fix" the "fuse". Here it is likely you will need to pull the machine away from the wall. The fuse is a black plastic button about the size of a penny. It sits in the center of the machine's backside. It faces up a bit and if you have clearance you can reach your arm over the machines top and press it "in" with a finger. It clicks when you do so. This part is a no brainer. The dyer will now start properly.
The hard part to all of this is pulling the machine away from the wall since like me you probably have it sitting on top of an Asko washer. Take your time. You can do it.

After cleaning the internal filter as well as the fan and checking the fuse it turned out it was an issue with too much water in the drain pan directly BEHIND the internal filter. I used a sponge to soak it up, put the filter/cover back on and my clothes are finally drying. Bonus though...taking off the fan and resetting it stopped an annoying squeaking! 

I had this problem where the dryer would only run if I held the button.

I removed the internal filter (the one you seldom clean that is only visible after you remove the bottom front panel and follow the pictured directions to turn the red handles). I cleaned the filter according to the pictogram, flushing it (a lot) with water in both directions. Make sure you're flushing the water into an area where you don't mind there being lint at the end.

That didn't fix it, but per Knash42's directions I sopped up the extra water in the area behind there the internal filter goes with a sponge. Took about 5 minutes and worked great.

I didn't have any luck holding down the button, but I also didn't have to access the back of the machine.

You may find that thers to much water in the dryer when you hold the start button in the drain pump will come on and drain the water out,then the dryer will work ok

There is a relay that has failed. it can be replaced. look for a rectangular part (plastic) on either a circuit board or mounted separately that has wires (or pcb traces) leading to or from the push button. The relay "holds in" the switch electrically when you release the switch. if you cannot find the relay, I can give you more details on how to add your own relay...just tell me how many wires to/from the switch.