Over time, your heater buttons will still light but the little light that reminds you it's "ON" will not light anymore (or will light sporadicaly)
This is caused by a bad connection. Over time. the solder breaks around the connection and becomes loose, thus, the light will work once in a while. It's very uncomon for LEDs to burn.
It's really simple to fix, just re-solder those connectors.
I admit that you got the controls out of the car before attempting that. If you cannot get them out, maybe you should not attempt to fix that and bring it to a shop with my instructions.
To get to the circuit board, you need to open the case. There are 4 tabs and 2 screws that you need to remove to get inside the case. There are 2 tabs on each side.
Once you removed the back of the case, time to get the circuit board out. There are 3 or 4 screws holding it (depends of the brand and country). Get those screws out and pull the board.
With the circuit board, locate the LED connectors. There are 2 pins for each LED
All you have to do is remove the old lead sitting there and resolder it. If you want, you can get new LEDs that got a different color (like blue or red)
Reinstall everything in place and enjoy!!!