I have a Kenmore Elite Gas Dryer in which the drum does not turn. After selecting the mode switch and depressing the start button, the dryer only hums with no turning of the drum.

Disconnect the dryer from its power source before servicing!

Inspect for a brokren drum belt. The humming noise is probably the motor running with no belt. You can easliy identify this by opening the dryer door, rotate the drum by hand by using a baffle as a gripping point. If the drum rotates freely, the belt is gone. If you feel a slight degree of resistance or drag, then you will have to go internal to inspect for the belt and associated rollers and idler pulley to make sure they are free rolling.