The aluminum tape is applied for grounding/shielding purposes: the unit will function without it as long as you re-apply the foam blocks that keep the cabinet cover from touching the internals. It would still be better to buy a roll of the tape and re-apply it as done on the original HDD, however: these recorders can be very sensitive to the slightest HDD glitch, including spurious electrical noise or poor grounding. Since you say this particular 640 came to you without the original HDD, you likely didn't get the foam cabinet-standoff blocks either: at a minimum, you should not connect the unit to power without those blocks. It is remarkably easy to short out and destroy a recorder motherboard by cabinet pressure on one of the internal metallic parts (HDD, DVD, PSU, etc).
Best practice would be to study one of your other Pioneers and apply fresh grounding tape and spacer blocks to the repaired unit to match.