What happens if you take heat away from an ice cube?
There isn't really going to be any spectacular event, from the macroscopic perspective of seeing matter changing phases such as when it melts, freezes, etc. The ice will remain ice. On the atomic level, however, there's still quite a bit going on.
Phases of matter are largely based on our qualitative perception of them; solids have a fixed volume and shape, liquids have a fixed volume but variable shape, and gases have variable shape and volume. These properties are largely based on the relationships taking place between the atoms and molecules that constitute the object in question.
In a solid, there are fairly strong bonds between many of the particles constituting the object, forcing them into a relatively static physical relationship. Another way to think about it is that anything that is solid, regardless of temperature, is "frozen" - it's just that things freeze at different temperatures because of their unique properties. Remember also that "cold" is a relative term.
When something is solid (aka frozen) its particles are pretty much stuck in place, but they still have some "wiggle room". This wiggling is heat, and it's how any solid is able to change to a liquid, or change temperature at all. If you remove heat from a solid system, that wiggling will reduce. So, basically, lowering the temperature of an ice cube will simply result in less movement at the molecular level.
The practical limit of this process is absolute zero, -273.15 Kelvin, the lowest possible temperature. Based on experimental data, there might be some pretty weird things that start to happen as the water approaches this temperature, but this would be a separate consideration.