Of course, plumbers fix furnaces. Think about the vast diversity of plumbing work that goes on behind the scenes at any given time in an average home. This includes not only pipes themselves but also sinks, toilets, bathtubs, water heaters, dishwashers and even sump pumps.
For many years now it has been standard practice for qualified plumbing professionals to complete tasks related to maintenance of heating equipment as well. While this traditionally was limited to gas-fired units exclusively (either natural gas or propane), changes in the natural gas industry have allowed plumbers with training and experience dealing with oil fired furnaces as well.
There are many different kinds of heating systems that your plumber might need to