There are four concepts here that you need to think about here:
1) Energy
2) Thermal energy
3) Temperature
4) Heat
Of these 4, only energy is conserved. Thermal energy can be converted to other forms of energy, such as chemical potential energy, so thermal energy is not conserved. Temperature is a relationship between thermal energy and entropy, it is not conserved. Heat is a transfer of thermal energy between two bodies based on a difference in temperature, so it is not conserved either.
When you have a room at room temperature with some ice in it the room has a higher temperature than the ice. Due to this temperature difference thermal energy can be transferred from the room to the ice, this is called heat. The thermal energy removed from the room will decrease its temperature. The thermal energy added to the ice will not increase its temperature, but will instead act to break the hydrogen bonds keeping the ice solid, causing melting. The resulting water will have more energy than the ice, although the temperature is the same. Only the total energy would have been conserved.