The fire would heat the room.  The fire is releasing chemical energy, from the fuel, into the room.  The energy is released as heat.  The room would therefore heat up further, as there is more heat-energy present in the room when the fire is lit.  To cool the room, you would need some mechanism to remove heat energy from the room, and a fire does not do this.

Blowing on hot food to cool it down works mainly by convection.  The hot food heats the air around it, and a hot air-layer is actually a pretty good insulator, serving to keep the food hot.  Admittedly, the hotter air does rise and takes some of the heat with it, so the food will cool down.  Blowing all the hot air away speeds this process up.  Using cooler air will increase the effect, but breath is cool enough compared to food we find 'hot'.  Similarly with cold food, you are blowing away a cold layer of air, and the air you blow from your lungs will be at body temperature - higher than a typical ambient room temperature.  So, blowing warm air to displace cold air causes the ice cream to melt faster.

To go back to the room question, it's instructive to think of an air conditioning system located entirely in the room.  It wouldn't cool the room, because the heat taken out of the room at one end of the conditioner would be put back into the room at the other end.  The net result would be no change.  This (and the convection reasons above) is why AC units need to have the heat-exchanger end located outside, where the heat can be disposed of without it going back into the room.

I agree with everything OO said, but would like to add a point about blowing on hot food of most kinds to cool it. The cool air you blow not only displaces the insulating layer of hot air, but also replaces the water-saturated hot air with cooler, drier air that becomes even less water-saturated as it warms up on contact with the hot food. This increases the net rate of evaporation from the moist, hot food, which increases the rate of cooling; in suitable circumstances it can increase it quite drastically.

This effect would be less important in dealing with say, chocolate, which is mainly fat, or hard sugar, but most kinds of food, such as meat, soup, or baked potato, contain quite a lot of free water.