There are two types of heat conduction in a fire (the third, convection, doesn’t apply in this case). They are radiation and conduction. Radiation is infrared light emitted by the fire, and is what you feel when sitting around the fire. Conduction is heat transferred via direct contact with the heat source. When you put the ice in the fire, it immediatly cools down what it touches. The heat of the ashes the ice is touching is tranferred, but the cooling down of the ashes prevents a constant energy transfer rate. After the initial heat transfer, there is less and less heat to be transfered. Ashes are also a bad heat conducter, and that is why people can walk on hot coals. Ice is also a bad heat conducter, and is why igloos retain heat. This creates a situation in which there is very limited heat conduction between the ice and fire. That leaves radiation. However, ice is mostly transparent and doen’t absorb much light. The radiation that ignites wood on fire goes right through the ice. I have seen this effect myself, and that is why people dump water on fires to put them out, not ice.