How do you remove a kitchen sink that was installed with a chaulk adhesive, without damaging the Formica countertop?

Remove the sink clips under the sink, then use a putty knife and carefully work it around the sink. If the formica is smooth and in good condition, I use a razor blade pushed firmly to the countertop and work slowly and steadily around the front and sides. Once you can break the front and sides free so that they will lift, work around the back side with a putty knife. Most plumbers use silicone, and not actual adhesive. If someone used construction adhesive, you may be replacing the formica (which is actually pretty easy, just messy).

so much depends on how much caulking was used.
if you only see a nice neat thin line of caulking, it will be more difficult than if you see a good thick bead.
i haven't removed one since i got my nice new oscillating tool but the tool comes with a flat blade that cuts into caulking just wonderfully.
you do need at least the thickness of the blade to get started.

If they got "generous" with the caulking, it may be inside the opening between the rim and the countertop. In that case, you have to be extra careful, because if you pull it up it will crack the formica also.

There are threaded fasteners under the sink. Lossen or remove them. Sink should pull up or out completely. You will need to clean the tof the counter AND the underside of the sinnk BEFORE re-installing however. Remove the faucet first so basin is the onlyn thing you see or have in the opening. You will need a basin wrench to losseen the nuts on the faucet. Perhaps a deep well socket for the wing nuts. You DO NOT need a great amount of pressure on the nuts.