No, you do not have to close your business for a video shoot. But, some precautions need to be taken when filming starts. If an interview, script reading, or any type of talking is being done for the video, every single person in the office needs to be quiet. Play the quiet game. Do whatever you need to do be silent and still.
All phones need to be turned on silent or turned off to avoid any noise. You will live without a few minutes of using your phone. No one needs you that badly. This is not the time to gather around the water cooler for office chat. If your A/C unit sounds like a freight train, then it too needs to be turned off. A few minutes of still air can be tolerated so that the video will be well produced without random, annoying noises. If the inside of the office or space is simply being filmed it is best that everyone does what they normally do and goes about their business. That sort of shooting does not require sound or the muting of every person in the office. Think about which route you want to take and prepare for what is to come.