The dash lights on my 1981 300D were not working at all when I bought the car this year (2004). I pulled out the instrument cluster and removed the rheostat (the white ceramic block on the lower left side of the cluster). I unscrewed the panels from the back of the instrument cluster to get to the front of the rheostat, because I thought it might have some attachment mechanism on the front, but I think that was unnecessary after all. Once the main screw was removed from the rheostat, I was able just to slide it off the posts on the main board with a little force (it's got a pretty good grip on those posts).

Once I had the rheostat off, I realized that it wasn't working because its copper parts were badly corroded. One option would have been to short the posts, like people describe here. But I wanted it to work right. So this is what I did:
1) Gently bend out the tabs which hold the rheostat assembly together.
2) Disassemble the rheostat parts. Be careful because there are a couple of little springs in there that you don't want to lose.
3) There's a central copper contact plate which is pressed against a long, spring-like resistor. As you turn the knob, the position of this contact moves further out or in along the resistor, changing the dash lighting brightness. There's also a copper stud (shaped like a rivet) at one end of the resistor. When the lights are turned 100% on, the central copper plate should make contact with this copper stud. On my rheostat, both of these copper parts were corroded (the stud had bright green oxidation, and I think the center plate was dark brown). The parts seemed too delicate to clean with a brush or steel wool, so I used a chemical solution, as follows.
4) Prepare a de-corrosion solution in a teflon pan: mix 1/4 cup baking soda in 2 cups hot water. Adding a few teaspoons of salt seems to help too. Place a piece of aluminum foil in the bottom of the pan.
5) Place the copper plate and the main part of the rheostat assembly in the solution, on top of the aluminum. Heat this solution and the parts -- I think I had it simmering, but it could be cooler than that if you want.
6) Keep parts and solution hot for a while. Stir occasionally with a wooden utensil. Chopsticks work well. Don't use metal utensils or your fingers. Add more baking soda and salt as needed (I think some of it gets used up in the reaction).
7) The corrosion will gradually disappear. It took 20-30 minutes for mine. You may want to rub the parts gently with a plastic brush, to give yourself something to do. My parts didn't get 100% shiny, but they were much improved, and the green went away. It's been a while since I studied corrison chemistry, but I think the baking soda/salt solution acts as an electrolyte, and the less-noble aluminum gets corroded to repair the more-noble copper.
8) Rinse the parts thoroughly with fresh water, and allow them to cool.
9) I had to gently buff the contact plate and the top of the copper stud with a wire brush or fine sandpaper to get a completely conductive surface back. Shiny copper is a good sign. It might have been possible to do this right at the start, but I appreciated having most of the corrosion removed before I went after the parts mechanically.
10) Carefully reassemble the rheostat. This is probably the hardest part. First I bent the metal bracket that holds the rheostat together until it was almost square again. Then I put the springs on the end of the shaft where they belong, pointed upward, and balanced the central plate on top of them. Then I lowered the main body of the rheostat down on top of this assembly. It helped if I had the shaft turned to the 100%-on position while I did this. Then I held these parts together while I slid the outer bracket back into position, to hold everything together. The first time I did this, I think I misaligned something, because the shaft wouldn't turn freely in the complete assembly. So I took it apart and put it back together again, making sure everything was where it belonged, and then it worked great.
11) If you have an ohm-meter, check that the resistance across the rheostat contacts goes from 0 to 10-15 ohms as you turn the knob through its range. If so, then you're in good shape. If you get an open circuit, then you've still got trouble. You may need to mechanically buff the parts some more, or you may have mis-assembled the rheostat. The central copper plate should be pushed by the springs against the copper stud or resistor wire on one side, and against an inner ring on the other side. Bridging these parts creates the necessary electrical circuit. If you can't get this to work, you may just need to solder wire across the two pieces of metal on the back of the rheostat, to force the circuit closed.
12) Reattach the rheostat to the instrument assembly and reinstall the instrument assembly in your car.

My rheostat now works approximately as it should. It's a little flickery in certain parts of its range, but once it's set in position, it seems to hold there, and the lights work as expected.

I hope this helps someone else.

Matthias

P.S. The green corrosion of copper is called verdigris, and here are the instructions that I based my procedure on: http://www.ehow.com/how_14456_maintain-brass-copper.html