We've got a Kenmore "Elite" dryer (actually made by Whirlpool) that stopped running when you release (let up) on the start button. After doing some research I found out that there is a $5 relay (every other place you try to buy it is going to be $35-50 each) that USUALLY fixes this problem.
Disclaimer: I take no responsibility for you spending $5 plus shipping on a part that doesn't fix the problem.
There are 2 of these relays in the control panel actually. One is the motor relay and the other is the heat relay. The motor relay was my problem. Supposedly if your dryer isn't heating up then it is the heat relay. The part number for both relays is POTTER & BRUMFIELD/TYCO T9AP1D52-48-03 and the Kenmore part number is 3405281. If you search around on the internet you'll come across a massive thread on Epinions dedicated to problems like this one. Sears will happily send someone out and charge you $200 to fix this problem unless you have an extended warranty which they will sell you for $200 for a single year. Honestly, I can't believe there hasn't been a class action lawsuit against Sears for these relays.
Just fix this yourself!
Disclaimer: I take no responsibility if you break your dryer even more.
On the sides of the front of the control panel you'll see these plastic moldings. Pull back on the top and remove them (they snap into place) exposing 2 screws on the bottom. Remove the screws and lift up on the front lower edge of the control panel CAREFULLY/SLOWLY and let it hang back on the soft plastic hinges. Make sure you don't snag on that resistor coming off the motherboard. You'll see the two identical black relays matching the spares you purchased (I bought 3 of them total for future use). The one on the right is for the Motor and on the left is for the Heat. Replace which ever one is the problem (be sure to hook the wires back up to the correct contacts). Put it back together and fire it up. Now, go take the $190 you didn't spend on fixing your dryer and save it for buying a non-Kenmore (do some research and avoid any manuf. that produces products for Sears completely out of spite).