I have a 2007 Chev Equinox. The radio and instrument cluster will not shut off when the ignition is turned off and the driver's door is opened. It will shut off after about 5 min. later. Any ideas?

Door sensor for what ? Door ajar light lit on the instrument cluster ?

What is it chiming ? Chimes for different reasons , headlamps on , seat belt not buckled , etc.... What warning light is lit on the instrument cluster ? if any . seat belt , air bag's etc...

Audible Warnings Description and Operation
The audible warnings alert the driver of a system concern or a critical vehicle condition. The following lists the audible warning priority and the pulse rate:
Fast rate chime (200 pulses per minute)
Medium rate chime (150 pulses per minute)
Slow rate chime (50 pulses per minute)
Single chime
Fasten Safety Belt Warning
The inflatable restraint sensing and diagnostic module (SDM) activates the fasten safety belt audible warning for 5-8 seconds. The fasten safety belt warning sounds and the fasten seat belt indicator illuminates when the following occurs:
• The ignition switch is in RUN, BULB TEST, or START position.
• The SDM detects that the driver's seat belt is not buckled (signal is low).
If the seat belt is buckled before the ignition is in RUN, BULB TEST, or START position, the chime does not sound. If the seat belt is buckled while the chime is sounding, the chime stops.
Key-In-Ignition Warning
The instrument panel cluster (IPC) activates the key-in-ignition audible warning. The key-in-ignition warning sounds when the following occurs:
• The ignition switch is in OFF, LOCK, or ACCY position.
• The body control module (BCM) determines that the driver's door is open (signal is low).
• The BCM determines that the key-in-ignition switch is closed (signal is low).
Lights On Warning
The instrument panel cluster (IPC) activates the lights on audible warning. The lights on warning sounds when the following occurs:
• The key is not in the ignition.
• The body control module (BCM) determines that the drivers door is open (signal is low).
• The BCM determines that the headlamp switch is in the park or head position.
Rear Parking Assist
The rear object sensor control module activates the audible warning. The audible warning sounds whenever any of the indicators are illuminated on the rear parking assist display. For a description of the rear parking assist system refer to Object Detection Description and Operation.
Turn Signal Reminder
The instrument panel cluster (IPC) sounds the turn signal reminder once per each flash of the turn signal lamps when:
• The ignition switch is in the RUN position.
• The turn signal switch is in either turn position, sensing, within 1 second, an open to B+ transition in the turn signal control circuit (left or right turn).
• The vehicle has traveled a distance of 1.2 km (0.75 mi) by counting pulses on the vehicle speed input circuit.
The IPC turns off the turn signal reminder when either the ignition switch is turned to the LOCK position or to the OFF position, or the turn signal switch returns to the OFF position, sensing, within 1 second, no transitions from open to B+ in either of the turn signal control circuits (left turn or right turn).
Additional Chimes
The following chimes will turn on when an associated indicator turns on in the instrument panel cluster (IPC):
• Low Fuel Indicator
• Low Oil Pressure Indicator
• OVER SPEED indicator
For a description of an indicator refer to Indicator/Warning Message Description and Operation.
For diagnosis of an indicator refer to Symptoms - Instrument Panel, Gages and Console.

Your have a problem with one of these systems , your best bet , take it to a qualified repair shop. They would do diagnostic's to find the problem .

Do any indicators illuminate when the chime sounds?
Go to Symptoms - Instrument Panel, Gages and Console
Go to Step 3
3
Turn the ignition OFF.
Remove the key from the ignition.
Turn the headlamp switch to OFF.
Close all of the vehicle doors.
Open one or more of the vehicle doors
Does the chime sound at all times regardless if the doors are open or closed?
Go to Step 15
Go to Step 4
4
Close all of the vehicle doors.
With a scan tool, observe the LF Door Ajar Switch parameter and the RF Door Ajar Switch parameter in the instrument panel cluster BCM/Inputs data list.
Do all of the Door Ajar Switch parameters display Door Closed?

With a scan tool, observe the Left Rear Door Ajar Sw. parameter and the Right Rear Door Sw. parameter in the instrument panel cluster BCM/Inputs data list.
Do all of the Rear Door Switch parameters display Closed?

This is a 21 step diagnostic trouble shooting flow chart .

There is no chime module ,there is a speaker inside the instrument cluster.
Door Ajar Chime
The door ajar chime feature is controlled by the smart junction box (SJB). The driver and passenger door ajar switches, and the liftgate are hardwired to the SJB. If any door or liftgate is sensed as being open, and the ignition switch is in RUN/START position, the SJB will process the input and sends a chime request to the instrument cluster over the controller area network (CAN).
Safety Belt Warning Chime
The safety belt chime feature is controlled by the restraints control module (RCM). The driver safety belt switch is hardwired to the RCM. The RCM communicates with the instrument cluster over a hardwire connection. If at any time during the first 6 seconds after the ignition switch is placed in RUN the driver safety belt is sensed as being unfastened, the RCM will process the input, and in turn, sends a chime request to the instrument cluster.
Have the vehicle checked for DTC'S - diagnostic trouble codes ,not OBD 2 codes , the check engine light has nothing to do with this .
Instrument Cluster Diagnostic Trouble Code (DTC) Index DTC Description Source Action B1342 Instrument Cluster is Defective Instrument Cluster INSTALL a new instrument cluster. REFER to Section 413-01 . B1352 Ignition Key-In Circuit Failure Instrument Cluster Go To Pinpoint Test B . B2477 Module Configuration Failure Instrument Cluster REFER to Section 413-01 .
For a complete list of instrument cluster DTCs, refer to Section 419-10 .


Smart Junction Box (SJB) Diagnostic Trouble Code (DTC) Index DTC Description Source Action B1342 SJB is Defective SJB INSTALL a new SJB. REFER to Section 419-10 . B1352 Ignition Key-In Circuit Failure SJB Go To Pinpoint Test B . B1577 Lamp Park Input Circuit Short to Battery SJB Go To Pinpoint Test E . C1167 Park Brake Actuator Assembly Switch Released Circuit Short to Ground SJB Go To Pinpoint Test F . Your best bet is to have a qualified repair shop diagnose this problem ,what ever it is .
Have the vehicle

The liftgate can be opened by operating the switch on the instrument panel. Battery voltage is supplied to the liftgate actuator through the body control module (BCM). The BCM will pulse the liftgate actuator when a release input is received through the key fob, or the instrument panel switch.
Not sure what your talking about . There is no sensor. there is a relay in the I/P fuse box LIFTGATE PCB relay . an a lift gate actuator motor inside the door . An there is a door ajar switch ,which is part of the latch mechanism. Do you have light lit on the instrument cluster - a picture of a little car with a door open , an maybe a chime sounding ?

Not

here is a copy of your owners manual the information is on page 391-396 https://cdn.dealereprocess.net/cdn/servicemanuals/chevrolet/2007-equinox.pdf

Its probably the gage itself. A quick stop by a Dealership and they can check the instrument cluster gage by gage. For free maybe! If not the gage, could be the engine temperature sensor.
As long as the engine is NOT overheating, you can still drive it.

It stay on and shuts off when it see a door open signal it should remain on when the ignition is switch off for approximately 5 minutes without a door signal when it sees a door switch signal then it shuts off this is normal on alot of GM product

The door ajar switch is part of the door latch.

Example the driver's door the wire from the door latch is Back w/ Yellow stripe wire ( switched to ground ) , the passenger front door is Black w/ Pink stripe wire.

These go to the instrument cluster, if the factory RKE is not installed. With the factory RKE, they are run to the BSM / VSM ( Body / vehicle security module ).

Is the problem the radio will not turn off when you open the driver's side door ( have to open the passenger to get the radio to shut off ) ?

check the fuse to the driver side window.

Try a window ,see if they still work too . RAP relay supply's power for radio windows etc. when key is turned off for ten mins or till a door is open .