Hi,
What is the make and model number of the refrigerator?
Can you hear if the evaporator fan inside the freezer compartment at the back behind a panel, is running?
It will stop when a door is open and start again when the door is shut (either door). It will run as long as the compressor motor is running. You may have to listen close to the fridge to hear it stop and start when you open and close a door.
Do not confuse this fan with the condenser fan which is outside the compartments near the compressor and can run whether the doors are open or shut and even if the compressor is stopped.
If the evap fan is not running, remove the panel at the back inside the freezer compartment (access through the front of the compartment) and check if the fan and evap units are "iced up" i.e. not just a coating of frost but chunks of ice, if so you have a defrost problem.