Filters in hot tubs need to be removed regularly so they can be cleaned. Some pool companies suggest buying two different filters so you can swap in a clean one while you are rinsing the used one. If you are using your spa regularly, cleaning filters about every two weeks is a good idea. More thorough cleanings every several months can help maintain your system.
The power to your hot tub should be on continuously, so your hot tub can monitor itself and move through running cycles during the day, depending on how you have it programmed. This constant power feed does not necessarily turn on the pump and does not pose a danger to you removing the filter. Because the filters were designed to be safely taken out, they are not attached to the electrical system of the tub.
If your pump is running because of a cleaning cycle or because you are currently using the hot tub, you should wait before removing the filter. Not only is the filter protecting the pump when the pump is working, but trying to remove the filter while the pump is working could be dangerous. Let the tub finish its cycle and switch off before taking out the filter. The power does not need to be disconnected for the hot tub to shut down.
For common cleanings, you should be able to rinse out your filter with a hose, aiming the stream of water at a 45-degree angle to wash away all trapped particles. For longer cleanings, you can soak your filter overnight in a hot tub filter cleaning compound and then rinse it out the next day.