From the looks of it, it should literally have changed the face of Earth. And we see the face of Earth a couple of times in DS9, unchanged. Doesn't really tell us much: the timetable of raising that new continent could have been decades or even centuries, and the project might still be perfectly on track. OTOH, it might have been a paper project all along, make-work for idle scientists and engineers, impossible in practice for political, emotional and possibly also ecological reasons (although I'd think eco-engineering would have been the major challenge in such a project, and the primary driving force). Timo Saloniemi
It would definitely displace a hell of a lot of water, but the scedule of the project and the technology at their disposal must allow them to deal with that. They've got matter transmutation, FTL drives, and weather control over the entire planet, a bit of water shouldn't be a problem.
They tried recruiting Picard on the project to raise a new continent on Earth which by the name would presumably be in the Atlantic Ocean. Certainly as mentioned, DS9's Earth was unchanged, but that doesn't mean much. I would imagine the concept of putting in a new continent would have to be done with layering and possibly would take years, easily covering the 8 or so years from season 4 of TNG to the end of DS9's run. Sure it would displace a ton of water. But I would think a society with transporters and entire weather control systems could get around that problem easy enough. I don't recall any mention it in any of the post TNG or DS9 novels I've read(though by all accounts I haven't read them all). Memory Beta says there's a TNG short story called Solace in Bloom that mentions the Dominion spying on it during the war, so that means at the very least they're still working on it. Given the incomplete nature of wikis, it might be worth asking this in the literature thread.
The Azores Plateau (aka The Dolphin Plateau) of the Mid-Atlantic Ridge would seem to be what the episode said was going to be the site of the Atlantis Project. Parts of this plateau are over a kilometre down, but the tallest bit is over 2 kilometres above the ocean surface, in the Azores Islands. The big problem I see is the blocking of the ocean current that flows from the Caribbean toward northern Europe. Without weather control, Europe is going to get very cold.
For all we know, water is what they are going to build this new continent out of! Either through replication of rock out of H2O, or by pumping seawater underground to adjust mantle temperatures... On the other hand, if the Federation has true climate control down pat, it probably makes heat transport by ocean currents rather redundant. Controlling heat flows at will on global scale would seem to be a necessary step in the climate control process. In TAS "Ambergris Element", tectonic adjustment was experimented upon, and supposedly later practically applied on a Federation planet as well. We don't know if the results on that other planet were as dramatic as on planet Argo, but we did witness the emergence of a large landmass from under the ocean there... Timo Saloniemi
The TMP novel mentions the Gibraltar dam, lowering the water level in the Mediterranean. Atr least at one time, Roddenberry saw Earth as engaging in geo-engineering, so they'd already have some practice with it. And the current would be rerouted by the mass of Atlantis - perhaps using solar mirrors to selectively heat sites in the ocean to induce the initial desired current? And perhaps some of the water was meant to be displaced into Death Valley, the sahara, or other inland seas? Or maybe they were going to beam up gigaliters to Luna, and this was a way to refill the bathtub a bit? If a permanent forcefield contained an atmosphere, and enough water and nitrogen was brought up, Luna could be terraformed into a shirt-sleeve environment. Although I think Mars and Venus would be better candidates - better able to hold on to the terraformed state without continuous power input at least. In in those cases, even with Trek tech, I'd use comets, instead, unless I needed a place to put excess seawater.