Dryer will not operate ,but it still heats even with the door open! Does not tumble or make any sound at all.
The start buttun doesn't feel as if it is making contact. Defective switch?

Sounds like the dryer motor centrifical end switch is stuck/contacts welded shut allowing the heater to be on when the motor is not turning. The replacement motor usually comes with a new end switch attached to the motor. You might check it out with an electric meter to be sure but I am pretty sure the motor/end switch combo is defective. The dryer motor end switch acts like a safety by not allowing the heater to operate when the motor does not turn preventing a fire or overheat condition.

Unplug the dry right away! It sounds like you have a shorted out heating element, and after the dryer overheated it blew the thermal fuse(s) which is why it won't run anymore. You will want to check for continuity from the heating element to ground, if you have continuity then the heating element is shorted out.

To fix this properly you should replace the heating element, thermal fuse(s), cycling thermostat, and check for any restrictions in the airflow (like a vent clogged with lent, or the dryer clogged with lint).