No, but it’s common that they do. Gravity furnaces weren’t terribly efficient so wrapping either the ductwork in asbestos, the furnace itself, or both, was something done regularly. This allowed the system to retain more heat. At the time, this greatly enhanced the efficiency of the furnaces but it wasn’t until later that we became aware of just how carcinogenic and dangerous asbestos could be.

Some pros may recommend covering a popcorn ceiling that tests positive for asbestos with beadboard ceiling planks or drywall. The cost to install a beadboard ceiling averages more than $5,000, but you likely will pay more than the average for the installer to deal with the asbestos safely. Installing a beadboard ceiling or new drywall will disturb the asbestos and release some microscopic fibers, so the installer will need to take precautions. 

No, you do not have to worry about new popcorn ceilings having asbestos. All residential building materials used since the mid-1980s stopped including asbestos, so new popcorn ceilings have no risk of asbestos. You can still create a textured drywall ceiling with a popcorn texture if you like, but new techniques have no risk of adding asbestos. 

If you know your popcorn ceiling has asbestos, you should not attempt to clean it. It simply is not possible to learn how to clean asbestos safely without potentially releasing microscopic fibers into the air that could lead to severe illness in the future. Avoid disturbing or cleaning the ceiling until the asbestos removal company can get rid of the popcorn ceiling.