What is the model number of the refrigerator?
Can you hear if the evaporator fan is running?
The evaporator fan will run as long as the compressor is running and both the doors are shut.It will stop when a door (either door) is opened
Without knowing the model number the following may not work but if the refrigerator/freezer have visible door switches located in the door jamb, when the compressor is running open the freezer door and then manually operate the door switch. If there are lights in the freezer compartment they should turn off but more importantly the evaporator fan which is located with the evaporator unit behind a panel inside the freezer compartment at the back will start running again. Check if you can hear it stop and start or feel the air from the vent when the switch is released and operated.
If there are no visible door switches (they could be concealed magnetic reed switches - depends on model), then listen closely and check if you can hear the fan stop and start when you open and close the door.
Do not confuse this fan with the condenser fan which is located beneath the compartments near the compressor and can run whether the doors are open or not.
If you can’t hear or feel the evap fan running at all, it may be iced up, it may be faulty, the door switches may be faulty or there is a problem with the power to the fan.
Remove the panel to reveal the evaporator unit and the fan.
If the fan is iced up then there may be either a problem with:
the defrost heater, located under the evap unit preventing sufficient ice from melting during the auto defrost mode, causing a build up of ice until it covers the fan
a blocked drain tube, located under the evap unit preventing the defrost meltwater from draining to the evap pan located below the compartments near the compressor resulting in a build up of ice
the control board auto defrost mode operation malfunction.
Also if the fan is iced up do not use a hair dryer on hot to melt the ice. Use it on the coolest setting for short periods at a time only, if at all - be patient, open the door and let it melt ;-), as the evap unit can be damaged if it gets too hot. Usually it is only thin aluminium and expensive to replace.
If the fan is not running and not iced up there may be a problem with the door switches. Check if the compartment lights (if also in freezer) go on and off when the switches are operated and released.
If the fan is running and not iced up etc there may be a problem with the air damper. This closes and opens to control the amount of cold air flowing into the refrigerator compartment from the freezer compartment.
All this depends on the model because some fridges have two evaporator units each with its’ own fan. One for each compartment but the principle of operation will be the same, except for the air damper connection between the two compartments. The evaporator fan is used to drag air across the freezing cold evaporator unit and blow it into the compartment(s) to cool them down to the set temperatures.
Hopefully this is of some help