Does the pump have to be on any time you’re using hot water?

If you want the benefits of having hot water delivered to the faucet/shower quickly, then yes. If the pump is not on, then your time to get hot water would be equal to the time it took before you installed the pump. Most users set the timer for the times they know they will need hot water so the pump will run in the morning and evening, but maybe not in the middle of the night or in the middle of the day if everyone is at work and school.