The Arctic Ocean does not drip downwards from the North Pole for the same reason that the Antarctic ocean does not pool downwards towards the South Pole: Earth's gravity pulls all surface objects down towards the core of the planet, not down from the northern "top" of the planet toward the southern "bottom."
Gravity is the force by which massive objects pull other objects with mass towards themselves. Gravity's effects range from the personal to the astronomical: our solar system is pulled into an orbit around the centre of the Milky Way Galaxy, Earth is held in orbit around the Sun by the Sun's gravity, the Moon is held in orbit around the Earth by the Earth's gravity, and individual objects, animals, and people are held against the surface of the Earth by the Earth's own gravity.
The Earth is nearly spheroid, and because its gravity comes from its mass rather than from outside "top-to-bottom" force, gravity tugs all objects on the surface inwards towards the core of the planet. Because of our perspective on the large surface, it can seem as though things are being pulled "downwards." In fact, however, things are actually pulled inwards, not simply downwards.
For this reason, from gravity's perspective there isn't much difference between an iceberg or an ocean in the Arctic, and one on the equator - or, for that matter, one in the Antarctic. Gravity does not pull a northern object down toward the south, but inward against the surface of the planet. Thus the ocean at the North Pole doesn't "drip" downwards because it is subject to the same sort of gravitational forces as objects elsewhere on the planet. If you stood at any point on the planet, including both poles, you would also perceive the same gravitational force, holding you down on the surface of the Earth.
There is one exception to this rule, though it is not confined to the North Pole. There is continuing debate (though increasingly not among environmental scientists) about whether global climate change is wholly a natural phenomenon or is partially a consequence of human industrial activity. However, there is no argument that gradual climate change is occurring (and, for that matter, has to some extent always occurred). As a result, glaciers across the planet are in retreat, and in both polar regions, ice sheets are receding. There is a slow dripping melt taking place in the Arctic as well as on Antarctica, as ice sheets pull back and the open sea expands. For one of the first times, the North Pole may soon be ice-free in summer, albeit only for a few days. The South Pole is covered by the dense ice of Antarctica, but on the edges of that continent, the ice is melting too, resulting in massive state-sized icebergs as enormous chunks of the ice shelf occasionally break off and float into the sea, where they take years to melt away entirely.