Guides I've Contributed To
I didn't remember to look inside the iPhone 4S for this piece of rubber but I believe they are present in both AT&T and Verizon iPhone 4's. The rubber wraps around the edge of the main logic board where the display and digitizer cables wrap around. It's not needed for proper operation but may protect the cables from long term flex and wear against the edge of the board.
Try a used plastic credit/gift card to apply gentle pressure to lift the battery out of the phone, the short edge of the card fits within the gap on the logic board side between the battery and video cables. You can also use it on the opposite side if there is room between the battery and iPhone frame. This has worked on all of the 4 and 4s models I've replaced screens on.
I offer an alternative image if interested.