How can I remove a porcelain sink from a granite counter?

A kitchen sink is usually relatively large and when made of porcelain it's probably pretty heavy. The putty knife is a good idea but you're fighting against not just the silicone but also the weight of the sink. Also make sure you're not fighting against the plumbing -- the drain or the faucet supply pipes could hamper your efforts.

Find a way to apply some lifting force to the sink while trying to cut the caulk with the putty knife. An assistant could push up on the bottom of the sink, or if the drain plumbing is removed, can lift the sink from above by grasping through the drain hole(s). Another possibility is to use a lever or jack of some kind. Take care with this approach. It sets up a "weakest link" challenge: if things go right the caulk will fail, but if things go wrong the sink may break or the floor of the cabinet may be damaged. Proceed with caution, increasing the force/leverage gradually.

Whichever method you choose for lifting the sink, try doing so at all four corners (one at a time). One of them may be adhered weaker than the others; once you get one area loose the rest will follow more easily.

Jack up the sink slowly. Take all of the plumbing out, get a car jack or any other type of small jack, put a piece of a 2x6/8 in between and push that thing up.

Once you get a little bit of upward tension you will see the areas that are stuck and they should have a few mm showing where you can cut the caulk. My go-tos are WD 40 and goo-gone. Give those a good 5-10 mins and then hit it with a really thin knife - I actually use an old tomato knife for cutting caulk or separating glue.

I have used the jack method on several stone types when I want to make sure not to damage. I for one wouldn't trying to demo the sink with it attached as one little piece that pops wrong could split a vein on the stone and take a little chunk out.