Copying virtual machines can be a little tricky – every different OS configuration can have its own quirks.
In the case of a debian installation with static ip addresses, replication is fairly straightforward, but it does have a thing or two that caught me off guard.
In addition to modifying the hostname (/etc/hostname) and ip address (/etc/network/interfaces), you have to update the mac address (/etc/udev/rules.d/z25_persistent-net.rules) in order to have your device recognized as eth0. Of course, different distros have different files, but just grep for your old mac, and you should be fine.
I suppose this change applies equally to any time you change network devices…