Thanks everyone for the input on how to remove the old finish. Removing the glove compartment door was easy. The old finish on it was so flaky the that I could mostly pick it off. The stubborn bits responded well to a hair drier.
I've already removed the old finish and applied wood grain filler to the bare wood.
My next question is, how do I get the other pieces off the dash? The bit on the passenger side offered me no clue as to how to get it off, and the piece on the driver's side looks even more daunting. But I'm committed. There must be a way to remove these wood pieces without taking out the entire dash!