Well, I'm not sure what year you've got. I don't know if it's the same on the brand new trucks, but take off the dash trim piece and you'll see four screw holding in the ashtray, unscrew and the ashtray is free.
Ditchdoc and Dodgegirl19,
Thank you both very much, don't mean to be a pain but are the pull tabs on the whole dash trim itself and are the tabs on certain corners? Just don't want to screw anything up!
I have a 2002 1500, Quad Cab SWB and had the same problem. The insert just slips out of the door that opens and then just pull the door out of the clips that hold it in place. Getting it back in takes a little longer but it is fairly easy.