There would be little reason to wear a compression dressing after a chin implant or osteotomy procedure. I only have my patients wear it overnite as its only purpose is to serve as a means to hold an ice pack in place. The next day it can be discarded. The implant is secured in place by the pocket into which it was placed and/or screw fixation. This is different than neck liposuction where longer periods of compression help to control swelling and provide some means of tissue adaptation.
I have not found that much for than a few days of compression is necessary with chin augmentation. A small bandage over the incision and a split piece of foam tape for 3-4 days is all that I have ever found necessary. The most important thing to do after chin implant surgery is to avoid any activity where you could get struck in the chin such as playing sports like basketball for 3-4 weeks. The recovery for chin augmentation is usually very simple.