Thermal conductors don't have to be hot, explain a situation where ice transfers heat

Temperature is all about relativity - what we define as hot is simply something that our skin feels as hotter than our own body.

Temperature is defined as the measure of the average random kinetic energy of the atoms in a substance - therefore the transfer of heat can only go from the warmer object to the cooler one. The process of transferring heat is literally the atoms in the warmer substance hitting the atoms of the cooler one and transferring some of that kinetic energy to them, resulting in the warmer object's particles slowing down and the cooler ones warming up - hence eventually, the two objects reach thermal equilibrium - that is to say, both of their atoms have the same kinetic energy.

Hence, to answer your question, a situation where ice transfers heat would be when it is in contact with something colder!
Let's think of an example - make oxygen is very cold, with a temperature of somewhere between 55 and 90 Kelvin and we have it in liquid form.
The freezing point of water (i.e. when it will be ice), is 0 celsius which is about 273 Kelvin.

Therefore, if we stick that about 273K block of ice into a pot of liquid oxygen at somewhere between 55 and 90K, it's going to transfer a lot of heat!