Heat causes the reaction between copper and oxygen to proceed faster.
The oxidation of copper happens naturally. If you look up the
The fire will not use up the oxygen before it can react because the copper is not being set on fire. The copper is in a bowl that is over the fire. So the bowl will heat up and transfer the heat to the copper. And since oxygen is in the air, that means that the oxygen is everywhere, including inside the bowl where it can react with the copper.