I have a 1994 Chrysler Concorde whose heat comes on by itself.

Check the electrical connector. If you unplug it and plug it back in and it works for a few days or weeks, there may be a light film of corrosion on the pins. The scratching action of plugging it back in will often grind through that corrosion. If you can make the display turn on and off by wiggling the plug, suspect broken solder connections on the circuit board inside. That's been a real common problem with tvs for decades and it can affect anything that was machine-soldered during assembly.

If you can't do anything to affect it, search for a replacement at one of the pick-your-own-parts salvage yards. If you live anywhere near Ohio and southern Georgia, there's a real nice chain of two dozen yards called "Pull-A-Part". Parts are very inexpensive so it won't cost much to buy one for testing. You can do an internet search to see how many of your model they have in each yard but they can't tell you options, colors, and which parts have already been removed.