As modern HPC computing platforms become increasingly heterogeneous, it is challenging for programmers to fully leverage the computation power of massive parallelism offered by such heterogeneity. Consequently, task-based runtime systems have been proposed as an intermediate layer to hide the complex heterogeneity from the application programmers. The core functionality of these systems is to realize efficient task-to-resource mapping in the form of Directed Acyclic Graph (DAG) scheduling. However, existing scheduling schemes face several drawbacks to determine task priorities due to the heavy reliance on domain knowledge or failure to efficiently exploit the interaction of application and hardware characteristics. In this paper, we propose INSPIRIT, an efficient and lightweight scheduling framework with adaptive priority designed for task-based runtime systems. INSPIRIT introduces two novel task attributes \textit{inspiring ability} and \textit{inspiring efficiency} for dictating scheduling, eliminating the need for application domain knowledge. In addition, INSPIRIT jointly considers runtime information such as ready tasks in worker queues to guide task scheduling. This approach exposes more performance opportunities in heterogeneous hardware at runtime while effectively reducing the overhead for adjusting task priorities. Our evaluation results demonstrate that INSPIRIT achieves superior performance compared to cutting edge scheduling schemes on both synthesized and real-world task DAGs.