It's a common misconception, backed by the PC manufacturers, that opening a PC will void the warranty. This is not true, as a PC (Personal Computer) by definition can be Personalized by adding (not replacing) new hardware, provided this is all you do. As long as you apply great care, and maybe document your steps (to be on the safe side), you shouldn't have any problems.
If you think this is too risky: generally, you can move the discs to the old PC, and Debian should largely boot as if nothing happened. If it does give you problems, boot it into single user mode (append "init=S" to the kernel arguments), and disable the services that act up. To connect the 2 PCs, unless you also have a hub or switch lying around, you'll need a
crossover cable.
One thing to keep in mind: if the machine you intend to transfer files from is as old as you suggest, it probably only has a 10Mbit card, which would mean a max of 1 MB/s transfer speed, or about 18 minutes per GB.