No, opening the door of a refrigerator in a closed room will not cool down the room faster compared to keeping it closed and running only a fan inside the room. In fact, opening the refrigerator door will likely make the room warmer rather than cooler.
Refrigerators work by removing heat from inside and expelling it outside. When you open the refrigerator door, warm air from the room enters the fridge, and the refrigerator has to work harder to cool down the new warm air. This additional heat released by the fridge's cooling process will counteract any cooling effect it has on the room.
On the other hand, running only a fan inside the room does not remove heat from the room; it simply circulates the air, providing a cooling sensation to the occupants by increasing evaporation from their skin. However, it doesn't actually lower the room's temperature; it just helps people in the room feel cooler by enhancing the body's natural cooling mechanism.
If you want to cool down a room, it's more efficient to keep the refrigerator door closed and use a proper air conditioning system or an actual room cooler to reduce the temperature. Alternatively, you can use ventilation techniques like opening windows and doors on opposite sides of the room to create a cross breeze, or use reflective shades to block out sunlight and prevent heat gain.