What is the model number of the refrigerator?
Check if the evap fan in the freezer compartment is operating.
It will run when the compressor motor is running and will stop when a door (either door) is opened.
Do not confuse this fan with the condenser fan which is underneath the compartments near the compressor motor and can run regardless of the position of the doors.
If you can't hear the evap fan, turn off the refrigerator and disconnect the power and then remove the back panel of the freezer compartment to view the evaporator unit and fan. Access to the panel is through the door at the front.
If the fan is stopped by being iced up, then there is a defrost problem e.g. faulty defrost heater, defrost thermostat, blocked drain line.
If the fan is not iced up, reconnect the power to the fridge and check if the fan operates by manually operating the door switch. This should turn off the light (if there is one) and turn on the fan by fooling the controls that the door is shut ;-)
If it doesn't operate it may be a faulty fan. The evaporator fan is used to blow air across the icy cold evap unit (0°F) and into the refrigerator compartment to cool it down. The freezer compartment also gets the air but the evap unit is there anyway which helps.