Moving all the files from one file system to another is not a problem. If just the files and permissions are the same, Linux will not care if the underlying hardware, filesystem type etc has changed. But Grub will, so you need to reinstall that.
If you like you can just make an image of your whole disk and convert it to a VirtualBox disk (using
VBoxManage convertfromraw mydisk.img mydisk.vdi). This will work without reinstalling Grub. But of course every other partition gets included too (which might make the file too big and unpractical). Gparted or the like can then be run from inside VirtualBox to edit the partitions.
Moving Linux to new hardware generally works fine. According to my experience the only thing needing some tweaking has been the X server configuration. But since you are running a server it's probably not even an issue. The new network adapter (as Linux sees it) will propably get a new number (for example eth1 in stead of eth0) so any explicit routing rules need to be updated (but you can tell udev to use old number, avoiding any problem, see this question).