Are dimples on my face from injuries permanent?

Not sure. It depends what they are from. Some people develop small adhesions connecting the skin to the underlying tissue. These can be divided through a small opening in the skin to release the scar. To prevent the skin from scarring back down, some fat grafting can be done with pretty good success. If you are talking more about small pits, then dermabrasion may help.
Yes. The dimples can be helped with fascial resurfacing - there are specific protocols with doctors who specialize in these with laser treatment for it is called fraxel check your local laser clinics.