Ignoring surface tension effects, and assuming a much larger and relatively fixed volume and density of supporting water compared to that of the melting iceberg (i.e. not an edge-case of huge iceberg floating in a very small amount of water) and no loss/gain of iceberg mass over time to evaporation (which can be non-trivial in dry polar air) or precipitation, the water level will actually BOTH increase & decrease over time depending on the RATE of melting which depends mostly on the temperature of the surrounding water and the amount of wetted surface the ice has in contact with it. For most icebergs, the wetted surface also changes significantly over time as its center of buoyancy changes--hence iceberg rolling and sheer surface break-offs.
We're also assuming the perfect case of relatively pure fresh-water ice and sea-water salinity. When changes in salinity and "dirty-ice" are factored-in, it gets even more complex.
It's actually a very non-trivial solution which, among other things, is why predicting iceberg lifetimes and danger to shipping lanes in the open seas is still very much an art.
In the end, the iceberg WILL melt and a net additional volume of water will be added to the sea, but if one could measure things that precisely the sea-level change relative to the melting process will fluctuate up and down over the lifetime of the berg.
This is a prime example of why, back in my high-school trivia team days, I learned to dread questions about science. The problem isn't so much that the writers of trivia questions get the answers wrong (they do, but only very rarely). The problem is that they pose questions they think are simple, but they use real world examples ("an iceberg"), instead of the Platonic ideals they meant to use ("an ideal spherical iceberg composed of pure ice floating in a sea of pure water, neglecting evaporation, on a series of rainless days..." -- actually, come to think of it, it's pretty damn hard to construct the Platonic iceberg!)
When answering a science trivia question you need to work hard to cut your train of thought off at the level of the "obvious" answer before you get too far down the line to computing the "more correct" answer.
That is actually an easier version of the same question. The anchor version contains all of the same issues plus an extra curveball. If the iceberg had a giant boulder resting on top, you'd have the exact same question.
