Solution

The temperature in the room will increase.

A refrigerator operates on the principle of a heat pump. It absorbs an amount of heat $Q_1$ from the cooled volume and rejects an amount of heat $Q_2$ to the surroundings (via the radiator on the back).
To accomplish this process, the refrigerator's compressor consumes electrical energy from the grid to perform mechanical work $A$. According to the law of conservation of energy, the heat rejected into the room is:
$$Q_2 = Q_1 + A$$

If the refrigerator door is left open, it begins to absorb heat $Q_1$ directly from the room air and rejects heat $Q_2$ right back into the same room.
The net amount of heat the room receives during each operating cycle of the refrigerator is:
$$\Delta Q = Q_2 - Q_1 = A$$

Since the compressor consumes electrical energy and performs work ($A > 0$), this work is completely dissipated and converted into the internal energy of the air in the room. A refrigerator with an open door effectively acts as a standard electrical space heater with a power output equal to the power consumption of its compressor.

P.S. However, all this only applies to boring factory models. I don't know about you, but I modified my fridge: all the exhaust heat from the compressor is dumped directly into the shower water tank. As a result, the water is always hot, no money is wasted on a water heater, and if you open the door, the fridge actually works like an air conditioner, pulling heat out of the room (it's a pity, of course, that all this is only in the palace of my dreams).