A multimeter is an essential tool for working with controls. It's as expected for a professional heating guy to have a meter as it is for a doctor to have a stethoscope. Digital electronic meters are very different from the old-fashioned analog items that were expensive, fragile and hard to read. For under $50, you can get a nice digital meter that will let you easily test for AC voltage and continuity. That's all you need for basic control circuit testing.
For a little more money, you can add more features, such as testing for amperage. And, of course, for a lot more money, you can add a lot more. But it's not at all necessary.