Shut off the power to the furnace. Clean the area around the hole -- up to 1/4 inch in size -- with a clean cloth. Wipe off dirt, debris and oil, or the mastic won't adhere to it properly.
Apply a liberal amount of mastic to the hole. You'll want to end up with a thick covering when you're finished. Use a brush, trowel or a caulking gun. Wear rubber gloves if you use your hands to apply it.
Press the mastic into the hole. Spread it out evenly so that it's at least 1 inch beyond the hole on all sides.
Shut off the furnace. Clean the area around the hole -- over 1/4 inch in size -- with a clean cloth. Wipe off dirt, debris and oil.
Apply a generous coating of mastic to the hole. Be sure to spread the mastic at least 1 inch past the edges of the hole.
Cut enough strips of mesh tape with a sharp pair of scissors to cover the hole in the furnace duct. The strips should be approximately 4 inches long.
Place the tape strips over the hole and press them into the mastic. Overlap the edges of the strips. The tape will add reinforcement to the repair.
Apply at least one more coating of mastic over the tape so that it's completely covered.