A blast furnace removes the oxygen from the iron ore(hematite), so why is hot air added in? It just produces more carbon dioxide, that we don't need.
The coke is added and heated, then reacts with the oxygen, so why is the hot air useful?
Actually, the air is needed to help continue providing the oxygen in the first part of the chemistry equation, the production of carbon monoxide:
2C + O2 (subscript) ---> 2CO
The air is hot because there is a need to preserve the thermal integrity of the system. If regular temperature air were used, it would slow the process down, so hot air is used to accelerate the chemical reactions going on inside the furnace. Remember, heat, such as from hot air, is one of the things that will tend to cause a chemical reaction to take place at a faster pace.
The production of carbon monoxide then reacts with the iron ore:
Fe2O3 (subscripts) + 3CO ---> 2Fe + 3CO2 (subscript)