I have a Kenmore Elite Gas Dryer, it will run for about ten minutes, then starts to beep, then shuts down. I have noticed that the exhaust pipe get very hot when this happens. I have taken it apart and vacuumed it all out, it continues to run and turn off.

These driers will do that when the lint builds up like concrete on the inside of the air ducts. You need to totally tear it down and scrape out and wash all of the parts except the motor, which you vacuum out. Halfway doing it will not work. This includes the exhaust vent, all the way, even the outlet on the side of the house if you have one. You can go to YouTube to see videos on how to take it apart, but do a very good job or it will fail again quickly. I would get a new belt and change it while you have it apart and get some white lithium grease and grease any bearings you find. You need to clean out a drier every two years, at least, according to the manual. All sensors work together by comparing temperature, so all need to be cleaned. You just unplug the wires from the sensors and remove ductwork that they are attached on and clean the entire thing with soap and water. Make sure you go all through the dryer and get all parts like the blower fan and housing, vacuum out motor well and clean exhaust. Mark all wires you unplug so you know where they go. This is an easier job than you would think. You need a meter to test any fuses that may be blown, which I strongly suspect is the case. You will see them on the schematic diagram.
See: Dryer vent cleaning prevents fires - YouTube and: Cleaning out a dryer with top lint cleanout - YouTube
Cleaning out a dryer: By a woman 2 of 2 - YouTube
Manuals: http://dl.dropbox.com/u/8010163/Kenm...%20Manuals.zip