Firstly, lets see how a normal refrigerator works (when door is closed):

Refrigerators work as heat pumps (but reverse). With the help of a compressor they absorb some heat (via evaporator) from inside the refrigerator compartment and reject some amount of heat to the room (via condenser).

In the language of thermodynamics the heat equations can be shown as:


Here,

Q2 is the heat absorbed from refrigerator compartment
Q1 is heat rejected to the room
W is the amount of energy required to run compressor

From first law of Thermodynamics:

Q1= W + Q2

or we can write the above equation as,

Q1 > Q2

In other words the heat rejected to room is more than the heat absorbed inside the refrigerator.

Now consider the case when door is open:

 

As the door is open the heat Q2 is absorbed from the entire room instead of just the refrigerator compartment. While the heat Q1 is rejected to the room just as before.

Now we have already derived that:
Q1 > Q2

That is, the heat rejected is more than heat absorbed in the room.

Hence there is a net heat addition in the room and the temperature of room would increase


Further reading: 

The net heat addition to the room is:
Q1-Q2

But

Q1 = W + Q2,   (from 1st law)

rearranging the terms,

Q1 – Q2 = W

i.e. the heat addition to the room is equal to the energy consumption of the compressor.

PS:

We are assuming for the sake of simplicity that:

  • There are no other energy interactions going on in the room, i.e. no heater, air conditioner or a stove is being operated in the room
  • The walls of the room are adiabatic, i.e the outside weather does not affect the room conditions
  • The compressor is 100% efficient with no irreversible losses in the form of heat.