Lets say your idea is indeed a solution, which its not by the way.
What do you think is needed to pump that volume of water from sea level all the way up to the top of the mountains?
Electrically driven water pumps. And guess what? It needs a hell of a lot of electricity to drive those water pumps to be able to pressurize and lift the water to the top of the mountain.
Here is a study by the University of California on the amount of electrical power required to move all that mass
http://cetulare.ucanr.edu/files/82040.pdf Now add to that weight of water which is 8.34lbs per gallon by the way, the static pressure pushing down on a column of water.
That equals to 0.5 psig (appr) per foot of water column. So if say a mountain is 4000 feet, which is not that high for a typical mountain, then you have 2000 psig of water head pressure to push against!
Do you have any idea hoe much electrical power is needed? Where is electricity coming from? Mostly fossil fuel plants which in turn causes a depletion in the ozone layer.
This will lead to yet even more polar caps melting that in turn causes sea levels to rise.