In confined multi-obstacle environments, generating feasible paths for continuum robots is challenging due to the need to avoid obstacles while considering the kinematic limitations of the robot. This paper deals with the path-planning algorithm for continuum robots in confined multi-obstacle environments to prevent their over-deformation. By modifying the tree expansion process of the Rapidly-exploring Random Tree Star (RRT*) algorithm, a path-planning algorithm called the continuum-RRT* algorithm herein is proposed to achieve fewer iterations and faster convergence as well as generating desired paths that adhere to the kinematic limitations of the continuum robots. Then path planning and path tracking are implemented on a tendon-driven four-section continuum robot to validate the effectiveness of the path-planning algorithm. The path-planning results show that the path generated by the algorithm indeed has fewer transitions, and the path generated by the algorithm is closer to the optimal path that satisfies the kinematic limitations of the continuum robot. Furthermore, path-tracking experiments validate the successful navigation of the continuum robot along the algorithm-generated path, exhibiting an error range of 2.51%–3.91%. This attests to the effectiveness of the proposed algorithm in meeting the navigation requirements of continuum robots.