when cells are running in clay, do you need to keep the computer on?

No, you don't need to keep your computer on while cells are running in Clay. Clay is a cloud-based service, so once you initiate the runs, they will continue processing on Clay's servers regardless of your computer's status. You just need to ensure you have the necessary hardware, software, and connectivity to access the service when you want to check your results. Source: Clay: Terms of Service