Gate stays open and does not close unless timer switch is turned off

This sounds like a signal/switch issue. Check the tail lift switch. It may be a button built into either the tailgate or the frame it sits in. It'll will probably be a spring loaded button that is pushed in when the lift gate is closed.
Also check the wire harness that feeds from the body to the lift gate. These flex every time you open and close the lift gate so the wires inside can wear out or snap over time.

Follow the steps in the manual if you have a manual and if you don't have one I recommend that you get one.

Has anyone suggested replacing the BCM (body control module)

Open gate and find where the wiring enters the gate (bottom of door) pull back the rubber insulator and check for broken wires or wires touching each other.

I hope this is an easy fix, but I doubt it!
Try to locate the switch, it should be a push button, that opens when you lift the rear gate, and closes when you close the rear gate.
If that switch is broken, it will stay on.
Also, the switch, usually made of plastic, could have fallen down inside the hole where it mounts, also causing it to stay on.
Perhaps it need adjustment to turn off the light.
If the lift gate has been sprung or is not exactly where it was when new, you might be able to adjust it somehow to work.
Like if the gate does not come close enough, you might glue or screw a piece of metal on to the contact surface to make it turn off the light.
My sad suspicion is, put a piece of tape over the light!
That is what I have had to do in a few cases.
Be blessed.

The switch in on the latch itself. When the striker goes into the latch it also trigger the small micro switch.
You can try something like WD40 and spray liberally into the latch and open close it 10 times or so to see if it frees it up.

Hello, it sounds likethe switch that controles the light is jaming, try pushing the sswith in and out firmly with your finger. If you get the same responce as you do when uou close the tail gate you have found the problem. The switch may need replacing.

The lift gate latch is at fault the switch contacts in the latch are worn out replace the liftgate latch and your problem is solved

after you program the door, you have to push "sel" and override the load until the next setpoint.

There is a switch in the lift gate latch that closes when the gate is closed and turns off the light, this switch is defective, replace it and the light will go off.