Inspect the furnace housing, duct work and vents to ensure the skunk is gone. Depending on your duct work, a professional might have to perform this task for you.
Remove the furnace filters from the furnace. Discard the filters into an outside trash.
Clean the furnace casing with a solution of detergent, vinegar and water. Combine 1 cup white vinegar, a gallon of water water and two to three squirts of dish soap in a bucket. Wipe down the entire surface of the furnace with a rag or sponge. Discard the water and rag or sponge after use.
Mix a cup of white vinegar with a cup of water in a spray bottle. Spray the solution around the furnace and into all of the furnace vents. Spray the mixture in the air throughout your home and on the furniture and carpets. This will help neutralize the odor in your home, as well as the duct work.
Install a new filter in the furnace.
Place dryer sheets in the furnace registers. The sheets will capture odors coming from the furnace and emit the smell of the dryer sheet instead.