We present TASC, a Task-Aware Shared Control framework for teleoperated manipulation that infers task-level user intent and provides assistance throughout the task. To support everyday tasks without predefined knowledge, TASC constructs an open-vocabulary interaction graph from visual input to represent functional object relationships, and infers user intent accordingly. A shared control policy then provides rotation assistance during both grasping and object interaction, guided by spatial constraints predicted by a vision-language model. Our method addresses two key challenges in general-purpose, long-horizon shared control: (1) understanding and inferring task-level user intent, and (2) generalizing assistance across diverse objects and tasks. Experiments in both simulation and the real world demonstrate that TASC improves task efficiency and reduces user input effort compared to prior methods. To the best of our knowledge, this is the first shared control framework that supports everyday manipulation tasks with zero-shot generalization. The code that supports our experiments is publicly available at https://github.com/fitz0401/tasc.