Automated task planning algorithms have been developed to help robots complete complex tasks that require multiple actions. Most of those algorithms have been developed for "closed worlds" assuming complete world knowledge is provided. However, the real world is generally open, and the robots frequently encounter unforeseen situations that can potentially break the planner's completeness. This paper introduces a novel algorithm (COWP) for open-world task planning and situation handling that dynamically augments the robot's action knowledge with task-oriented common sense. In particular, common sense is extracted from Large Language Models based on the current task at hand and robot skills. For systematic evaluations, we collected a dataset that includes 561 execution-time situations in a dining domain, where each situation corresponds to a state instance of a robot being potentially unable to complete a task using a solution that normally works. Experimental results show that our approach significantly outperforms competitive baselines from the literature in the success rate of service tasks. Additionally, we have demonstrated COWP using a mobile manipulator. The project website is available at: https://cowplanning.github.io/, where a more detailed version can also be found. This version has been accepted for publication in Autonomous Robots.