Well I would mostly dismiss the concern of the type:
Adding ice to the planet wouldn't remove energy from the planet, it would just add matter to the planet.
Sure, it increases the mass of the biosphere, but it would (in a cursory look) also decrease the temperature since the added water is lower energy than the average.
We need an additional first-level qualifier. I do believe (I'll check the numbers later) that the kinetic energy per unit mass for being in orbit far exceeds the melting heat of water.
This presents a bizarre quandary for people who seriously still want to ask the question. If you drop it from space, the gravitational energy of the ice will melt it and add even more heat. That will increase the temperature of the Earth, even initially.
So... we would drop it in slowly? This would involve something exotic, like a space elevator, but that will suffice as an plot device here. Ironically, lowering the ice would produce a tremendous amount of energy which could be harvested fairly directly from space elevator pulleys. So while the ice is intended to cool the Earth, it actually produces far more energy than the energy it subtracts from Earth. Clearly, using this extra energy on Earth would negate any benefit, but it could be safely used in space.
With those specifiers, we now have ice falling from the sky, but only a responsible altitude which leaves most of it as solid.
This would initially decrease the temperature of Earth. There is a certain "thermal inertia" time scale over which this applies. The heatup of the ocean itself is a major thing that delays temperature rise in the world we live in today.
Given enough time, however, the extra water will have an impact on Earth's greenhouse gases. In current studies, the water vapor positive feedback really accounts for the majority of the temperature rise. That causes more temperature rise for the addition of 1 unit of heat. Actually, the same feedback would apply when subtracting 1 unit of heat, so you get a multiplier of 2-3 because of that. That still doesn't change the temporary nature of the addition.
Regarding greenhouse gases, the critical question is how it would affect the water vapor in the atmosphere long-term. That would clearly be to make the Earth hotter, since adding more water just sounds like it would increase water vapor concentration. Come to think of it, this might even negate the initial cooling effect if it was vaporized before hitting the ocean!
There are a myriad of other feedback effects we could consider. If we compare albedo of different land types, we see that water has an extremely low albedo.
That would indicate that more water will cause the Earth to absorb more direct sunlight. It's hard for me to say whether this would have a larger or smaller effect than the water vapor increase, although I'm tempted to guess a larger effect. Anyway, they're both in the same direction.
I'm sure we could find an effect that cools the Earth due to the creeping shorelines. This is a notoriously complicated science. However, it looks like almost all long-term signs point to "hotter".
But for the near-term, that just means you have to add more ice at a faster rate. I doubt it would affect the conclusion by much. You would add more until flooding the Earth.