To be pedantic ... there will be an effect, because seawater is denser than freshwater. So the density of the iceberg is significantly less than the density of the seawater.
The icecube melting in a glass of water description only makes the level stay the same when the glass is full of a liquid and the "cube" is frozen material of the same stuff. This isn't the case with an iceberg, that has a zero or low salt content.
The force exerted up on the berg by the seawater is defined by the displaced mass of seawater:
(Force up on berg) = (Density of seawater) * (Volume of seawater displaced) * G
Meanwhile the (Mass of Berg) = (Density of berg) * (Volume of Berg)
and so (Force to hold berg floating) = (Mass of Berg) * G = (Density of berg) * (Volume of Berg) * G
Now if you assume that there are no air bubbles in the berg so that it is solid water-ice (if there are, it just makes everything harder but comes out the same in the end)
The forces must equal or the berg sinks or flies in the air, so:
(Force up on berg) = (Force to hold berg floating)
so
(Density of seawater) * (Volume of seawater displaced) * G = (Density of berg) * (Volume of Berg) * G
so
(Volume of Berg) = (Volume of seawater displaced) * (Density of seawater) / (Density of berg)
The density of seawater is about 3% more than freshwater, so this means:
(Volume of Berg when it melts) = (Volume of seawater displaced) * 1.03 / 1.00
So for ever 100 m^3 of seawater displaced by a berg when it is floating, will be replaced by 103 m^3 of freshwater, which adds to the sea volume.