Can a room be cooled by leaving the door of refrigerator open?

No, leaving the door of a refrigerator open will not cool a room, but will actually make the room hotter. A refrigerator cools its own interior by pumping heat to the heat exchange coils in the back (some models do not have exposed heat exchange coils, but the sides or back of the refrigerator have coils just underneath them, and they get hot). Those heat exchange coils pump heat into the room. Since the process is not 100% efficient, the amount of heat produced includes a certain amount of waste heat. As a result, pumping heat from one part of the room to another part of the room results in a net increase in heat.