A Memory Management developer on iOS is responsible for managing the allocation, use, and release of memory within an application's code. They ensure efficient use of memory resources to prevent leaks, crashes, and slow performance. This involves understanding and implementing concepts like Automatic Reference Counting (ARC), strong and weak references, and unowned references. They also use tools like Xcode’s memory graph debugger and leaks instrument to find and fix memory issues. They follow best practices to maintain the balance between memory usage and application performance, resulting in seamless user experience on iOS devices.