Is the burner safe to use when is has rusted over years from water dripping?

Rust on the outside shouldn't interfere with the coffee on the inside, though it will discolor the bottom of the glass pot. If you can, try to clean the rust off. Vinegar, baking soda and dish soap make a good cleanser and can remove rust. You can also try naval jelly or a wet piece of tinfoil to scrape the rust off. Then clean with a soapy sponge. In the future, try to maintain it by letting the coffee maker run with just water until you get ½ cup, then pour it on the burner and wipe. The combined heat from the burner and the water should loosen any accumulated debris.