Chronic wounds, such as diabetic, pressure, and venous ulcers, affect over 6.5 million patients in the United States alone and generate an annual cost exceeding \$25 billion. Despite this burden, chronic wound care remains a routine yet manual process performed exclusively by trained clinicians due to its critical safety demands. We envision a future in which robotics and automation support wound care to lower costs and enhance patient outcomes. This paper introduces an autonomous framework for one of the most fundamental yet challenging subtasks in wound redressing: adhesive tape manipulation. Specifically, we address two critical capabilities: tape initial detachment (TID) and secure tape placement. To handle the complex adhesive dynamics of detachment, we propose a force-feedback imitation learning approach trained from human teleoperation demonstrations. For tape placement, we develop a numerical trajectory optimization method based to ensure smooth adhesion and wrinkle-free application across diverse anatomical surfaces. We validate these methods through extensive experiments, demonstrating reliable performance in both quantitative evaluations and integrated wound redressing pipelines. Our results establish tape manipulation as an essential step toward practical robotic wound care automation.