It would be a good idea to close up the opening around the vent opening. If you don't, it is possible some of the forced air would go into the wall and not the room. How much would depend on the register fits on the opening.

If you give the opening a good vacuuming you should not have any dry wall dust problems.

I hope this is an interior wall.

use mastic tape and seal supply box to sheetrock
before installing supply grill
shop hvac supply house for hardcast brand #1402
mastic tape.
then remove all the other supply grills
and seal them.
here is a pic of finished install.

best of luck.

Someone needs to get the metal bent as close to the drywall as possible and then put some mesh up and finish it with mastic and seal the duct to the drywall. I would take a camera phone and take some pictures down the ductwork so you can see other issues and seal the rest of the ductwork. It makes a difference in comfort and efficiency.

Thanks for all the response!

BTW, this is interior wall. Just curious, what is the issue if this is not an interior wall?