The ice cube is always emitting, in an amount proportional to its temperature, area and emissivity. As long as those three things remain the same then so will the amount of radiation.
So far warming or cooling has not been mentioned .
An ice cube in space would quickly deplete its internal energy by emitting smaller and smaller amounts of radiation as its temperature decreased until it reached the same 'temperature' as space.
An ice cube in a minus20 freezer would cool to minus20 and then continue to radiate the amount proportional to minus20.
An ice cube in a 5C fridge would still radiate according to its temperature but it would now be absorbing more radiation from its surroundings than it was emitting. It is now warming rather than cooling.
An ice cube in a 25C environment warms even faster because there is now a large disparity between incoming absorbed radiation and outgoing emitted radiation.
An object radiates in proportion to its temperature and emissivity. An object warms or cools according to the net transfer of energy between the environment and object.
so sure, let's go there, so will the ice warm the coffee? Nope.
And yet you believe the colder atmosphere warms the hot surface. Warms the surface to almost two times that of the sun's incoming radiation. And you think I'm on the right side of the stupid IQ meter.