1998 GMC Sierra radio and display stays on all the time

First of all...does your truck have the retained accessory power (RAP)? If it does, the radio would always stay on with the key off until you open the drivers door. If it does have RAP and the door pin on the drivers door has went bad, then the truck doesn't know that you opened the door and continues to allow accessory power (although it would still cut off after about 20 minutes). If the interior lights come on when you open the door, then the pin is ok. If the lights come on, it could be the RAP relay. If it has gone bad, it is possible it is stuck supplying power and needs replacing. The relay should be labeled in the fuse panel, and there should be other relays in the panel with the same part number. You can swap a relay for the RAP relay and see if it fixes the problem...if it does fix it swap the relays back and replace the RAP relay. If the truck doesn't have RAP then it is possible the problem is in the radio..if you have a friend with a GM car or truck close to the same year you could swap radios and see if that fixes your problem...should be pretty simple. pull the cover off the dash, it just clips on, take out a couple screws that hold in the radio, unplug it and plug in the other radio. sorry so long winded...hope this helps you somewhat.

Is it the factory radio or an aftermarket one?