The tiled backsplash usually sits just off the countertop by an 1/8th inch or so and the gap is caulked. The caulk can be cut away and the countertop should be able to be removed without damaging the backsplash. That's in a perfect world. Some installers set the bottom course of tile right on the countertop. Less clearance. Depending on how the countertop is attached to the cabinets below, that can affect things. The shape of the counter. A straight run or and "L" shaped? Are the ends open or captured? How the countertop is constructed? A slab of stone? Or a tiled-in mud bed? Sometimes the countertop needs to be sectioned out in pieces. In general, the backsplash can usually be preserved, but getting the countertop out might be ugly, especially if it's piece by piece. So..."it depends".

I currently have top mount sink and it does not have to be replaced. I was just thinking that a particular faucet that I may like may not fit the current sink but would fit a new sink. You guys make an excellent point, however, because I would prefer an under mount sink w/ new countertops.

Photos would help. Your best solution would be to take your cabinet door and a sample of the floor with you when you shop for counters. What type of countertop are you looking to do?

Would help to know what material the new counters will be. Eased edges are not highly recommended for certain materials, although I do have them on a not recommended material. And FYI,, it's not just the edges but the corners inside and out to be concerned with. You might try asking this on the kitchen forum - this commonly comes up there and you'll get the good the bad and the ugly on various choices