Will it hurt my DC workers' compensation case to look for another job or take another job after my injury heals?
Not at all. The way it works is your weekly workers compensation benefits should continue until you are released by your doctor to return to your regular job or you return to another job making more money than you were when you got hurt.
If you have a permanent injury to an arm, leg, hand, foot, etc. (sometimes called a scheduled loss or permanent partial disability based on the schedule) you can get those benefits after you reach maximum medical improvement whether you are working for the same employer or not.
Future medical treatment and ultimately settlement will not be affected by a job change, either.