Yikes, I read a whole article on how messed up ducts make your heating system work twice as hard for half the effect. One clear sign is when some rooms stay much hotter or colder than others, no matter what you do with the thermostat. You might also hear rattling or whistling from the vents because the air is fighting to get through. The article mentioned blocked or bent ducts can build up pressure and cause drafts in strange spots. It's basically like paying to heat the inside of your walls instead of your home. Some folks really overestimate their skills with a roll of duct tape.
You won't believe how bad it gets when ducts are crushed or disconnected. Rooms farthest from the furnace stay freezing cold no matter what. I've seen systems where the noise from loose metal sounds like a haunted house. It's a total waste of money on heating bills.