This was a unique problem. The complaint on this Whirlpool dryer was that the drum would not turn, but the heating element was on. Right away this is a hint as to where the problem was. The element should never be able to be on if the drum is not turning.
So I pulled the back off and removed the element. When I flipped it over it was easy to see what the problem was. The plate that holds the element has warped which was causing the element to ground out. This means that anytime the dryer is on, it will be on and would not cycle off.
Replaced the element and the open safety thermal fuse and all is well.
EDIT: Oct 24, 2011 Here is another one. See how the center support warped up enough to touch the element.