Depends on the "surveillance camera".
There are many manufacturers that connect toe camera to a digital video recorder or network video recorder. No need to take the camera down - but connection to the DVR/NVR either using a smartphone, tablet or computer with a compatible client, network access and the correct userID/password - OR - keyboard, mouse, monitor connected to the DVR/NVR is needed. I use Swann systems. This video can also be sent to a storage server somewhere in the internet for remote access.
There are many "surveillance cameras" that are connected to the internet and send their captured video to a storage server somewhere and there is no DVR/NVR storage. Ring, and many others make these. I used to have a Ring Doorbell - but discontinued the service when we moved. I am not a big fan of the monthly subscription, but understand why it is needed - video takes up a tone of space and someone/something needs to pay for the servers and hard drives.
Some people need only a single "surveillance camera"... maybe two. In this case, rather than online or NVR/DVR storage, the camera stores video locally to flash memory like SDHC or miniSD cards. In this case, typically, swap the memory card with an empty one... then put the card in a computer. No need to unmount the camera.
Usually, the only time to dismount the camera is when the camera is advertised as "wireless" and the battery needs to be recharged. In addition to the power needed just to run the camera, the wifi connection needs power, too. Some cameras allow for quick battery trade-in (then the depleted battery gets recharged). I am not a fan of these "wireless" cameras - some are... I suggest at least two cameras are purchased for each individual camera location: one camera is in use while the other is recharging.
There are some low-end cameras that have built-in memory (not replaceable SD memory), that might need to be dismounted to get to their video (usually with a USB connection to a computer). I strongly suggest staying away from these sorts of fixed memory devices, but the use-case (and size) may not allow use of conventional removable memory cards.