Task planning systems have been developed to help robots use human knowledge (about actions) to complete long-horizon tasks. Most of them have been developed for "closed worlds" while assuming the robot is provided with complete world knowledge. However, the real world is generally open, and the robots frequently encounter unforeseen situations that can potentially break the planner's completeness. Could we leverage the recent advances on pre-trained Large Language Models (LLMs) to enable classical planning systems to deal with novel situations? This paper introduces a novel framework, called COWP, for open-world task planning and situation handling. COWP dynamically augments the robot's action knowledge, including the preconditions and effects of actions, with task-oriented commonsense knowledge. COWP embraces the openness from LLMs, and is grounded to specific domains via action knowledge. For systematic evaluations, we collected a dataset that includes 1,085 execution-time situations. Each situation corresponds to a state instance wherein a robot is potentially unable to complete a task using a solution that normally works. Experimental results show that our approach outperforms competitive baselines from the literature in the success rate of service tasks. Additionally, we have demonstrated COWP using a mobile manipulator. Supplementary materials are available at: https://cowplanning.github.io/