What are the implications of driving a drywall screw too far into the sheet?
The paper is a key part of the drywall structure. Just as when you cut the paper on the drywall it's easy to snap, when you screw past the paper it's easy to blow out the back. Plaster and paper are a lot like concrete and rebar, the plaster based core of drywall resists compression, while the paper resists tension.

It's not essential to pull the old screws, the damage has already been done and will be mudded over. But you should add a second screw a few inches away anywhere you went too deep. This is a critical repair for drywall on the ceiling, and a very good idea for drywall mounted on the wall. I wouldn't go through any added effort to tear down the drywall, since it's still perfectly good. Just add the extra screws and be happy you caught the problem before experiencing a collapse.



Update: From your new photos, those screws are too deep. Once the paper has been torn, you lose strength at that screw location. You should never be able to see plaster around the screw head, but you also should never be able to run a flat edge over the drywall and feel the screw head above the drywall. It's a fine line to walk, but a professional drywaller should be able to walk it with ease (we train amateurs to do this in under a hour).

Also, from those photos, it doesn't appear that the installers are using a drywall bit (we refer to them as mushroom bits because of their shape). They leave a distinct ring around each screw, making it easy to countersink the proper depth, and preventing you from going deeper (the bit will cam out when it hits the paper).