Metal is a better conductor of heat than water. Leaving the spoon in will allow the heat to escape the cup through the spoon. The rate depends upon the size and substance of the spoon.
Stirring the tea will also cause the tea to cool. The tea will at first increase in temperature as the molecules gain temperature in movement. However, more of the tea will be exposed to the air, causing the tea to release more heat than is generated by the stirring.
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