Chronic wounds, including diabetic ulcers, pressure ulcers, and ulcers secondary to venous hypertension, affects more than 6.5 million patients and a yearly cost of more than $25 billion in the United States alone. Chronic wound treatment is currently a manual process, and we envision a future where robotics and automation will aid in this treatment to reduce cost and improve patient care. In this work, we present the development of the first robotic system for wound dressing removal which is reported to be the worst aspect of living with chronic wounds. Our method leverages differentiable physics-based simulation to perform gradient-based Model Predictive Control (MPC) for optimized trajectory planning. By integrating fracture mechanics of adhesion, we are able to model the peeling effect inherent to dressing adhesion. The system is further guided by carefully designed objective functions that promote both efficient and safe control, reducing the risk of tissue damage. We validated the efficacy of our approach through a series of experiments conducted on both synthetic skin phantoms and real human subjects. Our results demonstrate the system's ability to achieve precise and safe dressing removal trajectories, offering a promising solution for automating this essential healthcare procedure.