How can I disassemble an iPhone 4 battery without damaging it?
Knowing Apple it might very well be a "tamper proof" firmware/hardware mechanism that will shut down the board when battery is disconnected from it. I seem to remember some talk of this when researching laptop batteries and replacing the Li-Ion cells (needing to keep power applied whilst swapping to prevent controller board shutdown) though I am by no means certain.
Google will probably throw some light on the matter if you use the correct incantation.  

Edit - had a quick look and managed to find this which seems to confirm this protection may indeed be present in many laptop batteries (2nd paragraph)
If you try again with another battery I would maybe follow the the procedure described regarding applying a secondary source through resistor to keep power present at all times.