Usually, No. You need not unmount your drive for data recovery. However, if you are a technician knowing the techniques of unmounting the drive, you can do it.

We usually require the hard drive to recover data. But, that doesn’t mean we handle the discs only. You can bring your system to us and we’ll get the drive out ourselves. You need not bother about whether the drive will be required or something else.

In case of laptops and some other devices, it is better if you send the whole system. If you try to get the drive out, you may damage the media more. So, it is always better to send the whole media.