Melting sea ice or ice shelves can indeed change sea level. It turns out that I was probably the first person to compute by how much the sea ice can do so, and there's a story for tomorrow about why I wasn't the person to publish this in the scientific literature even though I had the answer more than a decade before the next person to look at the problem.
The first peer-reviewed investigation was published by Peter D. Noerdlinger and Kay R. Brower, in The Geophysical Journal International, 170, pp. 145-150, 2007 The melting of floating ice raises the ocean level. The DOI is 10.1111/j.1365-246X.2007.03472.x You can get the authors' copy at the linked title. They included a simple experimental demonstration as well, which I hadn't done. I also didn't think about the ice shelf contribution, which turns out to be 10 times larger than sea ice's. Oh well.
The wrong answer on this question is to say that a floating body displaces its own mass, so when it melts, the water level is unchanged. Now, as this is partly quoting Archimedes and he was an awfully bright guy, there's at least good company.
The reason it fails (and, by the way, it isn't clear that Archimedes didn't know about this) is that what is melting when we melt sea ice or ice shelves is not the same stuff as what it is floating in -- sea water. Sea water is salty, about 3.5% salt. Sea ice is fairly fresh, about 0.5% salt. And ice shelves are completely fresh. If we were melting ice shelf in to fresh water, the level would indeed not change. You can test this with a glass of water and some ice cubes. To find just what happens when you melt frozen stuff that's floating in a liquid, you have to do the math. I've got it in my Sea Level Change FAQ. The result is, sea level rises slightly. I found a few millimeters (about 4) for sea ice. Noerdlinger and Brower found a few centimeters (also about 4) for ice shelves. It isn't much, but it isn't exactly zero.
A way to think about what happens, minus most of the math, is to envision a block of ice floating in the ocean. The density of ice is lower than ocean water -- about 917 kg per cubic meter for ice, versus 1028 kg per cubic meter for ocean water. The difference is why '90'% of an ice berg is below the surface. Conversely, 10% is above the surface. It's actually (1028-917)/1028 above the surface, 10.8%. Now melt the fresh ice, but don't let it spread out or mix with the ocean. That gives us a material with density near 1000 kg per cubic meter. That blob, for the same reason that the ice was floating in the first place, sits (1028 - 1000)/1028 of itself above the water level -- 2.7%. That bit sticking above water level means that melting such ice does contribute to sea level change. If the density of the stuff you melt is different than the stuff it is floating in, you can indeed have a rise (or fall, if the melt is denser than what it floats in) in fluid level.
This example is also why I, in particular, and scientists more generally, want you to 'show me the math'. The thing is, when I first wrote the sea level FAQ, I made the same error as everybody else was making. It was an early commentator (Rick Chappell) who told me I was wrong (he didn't have the math, but did have the above principle) that prompted me to work out the math in detail. I was doing show just to show him in detail that he was wrong. In fact, at the end of my calculation, I'd shown myself that I was wrong. So updated the FAQ and my thanks to Rick. Merely asserting principles would not have changed anybody's mind. The problem being that, although a principle might be true (if I jump, the earth moves the other way), the effect might be too tiny to worry about. It's only when we do the math, get quantitative, that we can decide whether something is too tiny or not.
A slightly different example, prompted by this point going back to ancient Greece. There is an Aristotelean (or at least his time) principle that "Nature abhors a vacuum." Now, for many purposes in daily life, this is not a bad principle. If you try making a vacuum, you'll need to do things to shield it against nature trying to fill it back up. On the other hand, it was also an argument that there were no such things as atoms. The argument being, if there were atoms (discrete bits of matter) then there would be gaps between the atoms. But those gaps would be vaccum -- and 'nature abhors vacuum'. Now that we know that there are indeed atoms, the principle applies, but it is directed to how nature responds, and in which cases.