Naaah!
You should shut down, connect the hard disk to a read only interface (it's a special IDE or SATA, or USB, etc... interface that doesn't allow any writes, something like this: http://www.forensic-computers.com/handBridges.php ) and do an exact dupe with DD.
You may do it to another hard drive, or you may do it to a disk image.
Then, store in a profer and totally safe place that hard disk, is the original proof without any tampering!
Later, you can plug that cloned disk, or image in your forensic computer. If it's a disk, you should plug it through a read only interface, and if you are going to work with an image, mount it 'read only'.
Then you can work on it, again and again without changing any data...
FYI, there are "hacked" systems images on the internet for practice, so you can do forensics "at home"...
PS: What about the hacked system bringed down? if I think that system is compromised, I wouldnt leave it connected, I would put a new hard disk there, and restore a backup or put a new server in production until the forensics finish...