As it turns out, the adhesives used on standard plastic duct tape dries out quickly and ceases to hold. If you want to use tape, choose butyl tape, foil tape, or fiberglass tape. Look for one that is approved for use on hot surfaces. Even a better idea is to combine fiberglass tape with a layer of mastic or a mechanical fastner. Smearing on mastic can be a messy job, but it is very effective in stopping air leaks.
Openings frequently occur at the joints of duct segments.
By applying generous amounts of mastic around all of the joints, you can stop the flows of air out from the supply ducts and into the returns. In the photograph above, a weatherization specialist is applying mastic with a brush to the joints of a supply duct. By the way, you need to know that not all leaks will be easy to reach. A hole or gap in an air duct deep in your wall may be quite costly to mend. Since the wall cavities in your house may be partially or entirely open to your attic, crawlspace, basement, or garage, sealing those openings could be very productive of energy savings.