2006 Hummer H3 radio, windows and wipers wont turn off when

More than likely it is a door jamb switch that is stuck open. What I would do is go around the vehicle, cycling it each time, and see which doors disable the accessories when open, and which don't. These door jamb switches are notorious for water intrusion, the terminals may be bad enough that they need replacement. On the door that dont disable accessories, remove the jamb switch(T-20 torx) and short the terminals with a paper clip/staple or jumper wire. If it disables, replace the switch. If not there is a break in the wire at the terminal crimp, or a bad terminal.

It is not too common, but also not impossible to have five bad jamb switches. Try the removing of one or two and shorting the connection together, like mentioned previously. It is a simple test and will determine if the circuit is functional. You could also use an ohmmeter or continuity tester, and bench test a switch. It would have continuity when the switch is at rest, open when compressed.
In addition, these are all inputs to the BCM(body control module). It supplies a common power internally to all switches, but on 5 seperate wires. The drivers switch, and the left rear are grounded through G300, and the passenger front/rear and endgate ground at G320. With this being said, the chance of having several circuit faults, or two open grounds, is more rare than 5 bad switches. About the only other possibility would be a faulty BCM.

Do the dome lamps come on when you open a door? Prior to saying no, check the status of the override switch. Have you tried the test that I recommended in previous posts, you have said nothing about results.