What is the sink made of? If it's stainless steel, these are usually just held in place with several bolts that go through clips and hold the sink tight to the counter. If you lie on your back under the sink, you should be able to see these bolts and loosen them. However, you can't really lift up the sink an inch without disturbing the drain connection, and, unless the water supplies are connected with flexible lines, those would have to be disconnected, too. After you've gone to all the trouble of loosening the bolts, you might as well take it out temporarily.

Also, what are you putting on top of the counter? If it is rigid, and you don't completely remove the sink, you won't be able to get the covering in place without splitting it in two pieces at the sink. I'd be interested in more details about what you are trying to accomplish because your fix may turn out to have more problems than whatever you're dealing with now.

OK, thanks for your answer, kudzu9. First I'll give you the details of which you asked, and then I'll tell you another thought I have.
1. The sink is stainless steel. I looked at the underside and the bottom of the sink is some blackish material, don't know what it is.
2. There is a garbage disposal attached to the pipe at the bottom.
3. It has a flexible hose somewhere that I see, I'm not sure exactly which it is attached to, I think it might be to the dishwasher.
4. I was thinking to put a flexible liner and then polyurethane over that. I know it sounds crazy, but that's kind of what I wanted to do. I did it partially to some of the counter, but realize I have to get under the lip of the sink for it to be really good.

Now ... if that is impossible (too hard for me to do by picking up the sink slightly), I have another question. I am hoping to replace the faucet situation. I have a faucet assembly now with three holes I suppose. One for the faucet and two for the handles. I want to replace that with a pull-out spray faucet, I suppose with only one hole, which would mean I'd have to take out the entire first assembly including the base for the faucets.

Let's start with your comments on this project as I've described it before I go on with my other ideas for the sink, thanks!

1. The black material is sound deadening to make the sink sound less tinny.

2. You're going to have to disconnect the drain line from the disposal before lifting out the sink.

3. The flexible hose is most likely for the dishwasher.

4. Not sure what you're talking about there.

Some of the single hole, all-in-one type faucets some with a trim plate to cover the extra holes. Read the box for details or check with the manufacturer if one can be ordered. Yes, you'd typically replace the old faucet with the new.

All of this being taken into consideration, I probably should change the faucet first before I start lifting the sink in order to put some material under the lip. OR ... I was thinking to put tile on the counter instead of the material I was planning to do, but the problem I would have is figuring how to fill in the little space that will occur between the edge of the lip on the counter and the tile (that is without lifting the sink).
Since you probably don't understand this, because I'm not being very explicit, I'll start by saying that I wanted to apply a surface material like a vinyl liner and then polyurethane over it to give it a hard finish on the countertops, but I wanted to get the liner material under the lip of the sink. However, another possibility would be for me to tile the top, problem is I'd have that little gap between the tile and the edge of the sink and I"m not sure how to fill that in, with what material, plus I don't know how to cut tiles.

Normally, if you wanted to tile a counter, you would have the sink removed, tile the counter so that the tiles go right up to the edge of the sink opening, and then set the sink back in. You would first lay a bead of silicone caulk all around the opening and then set the sink in place with the lip on top of the tiles. This would seal everything and you wouldn't have a gap. Another advantage of taking the sink out is that you can uninstall the old faucet and install the new one much more easily if you are not trying to do it from underneath.

It's relatively easy to cut tiles with a manual tile cutter, or even better, you can cut them with a diamond-bladed wet saw that you can rent for a modest amount. However, I'm not a big fan of tile on kitchen counters because it's hard to keep the grout lines looking clean over time.

Finally, I have another question: What is the material you would be trying to adhere tiles to?