So, the radio has three things that it uses to know when to be on or off.
1. Switched accessory power.
2. Starter interrupt.
3. Data buss to tell it when a door is opened.
If accessory is at +12V and Start is at ground, the radio will operate.
If accessory is at +12V and Start is at +12V, the radio will be off.
If accessory is at ground and Start is ground, the radio will turn off after ten minutes or after a door opens, whichever happens first.
Accessory should be at 12V anytime is key is in the accessory or the run position, and at (or near) ground any other time.
Start should be at 12V when the key is in the start position, and at (or near) ground any other time.
Time to get the meter and see what's going on. If the accessory and the Start line are doing what they are supposed to do, then the headunit itself is bad.