A failed electronic control board, faulty user interface, bad fan switch, wiring failure or failed fan motor can keep a range hood fan from running.
Unplug the range hood and check the wire harness connections between the control and the fan motor. Reconnect any loose wires and repair any damaged wires. With the range hood still unplugged, check the electronic control board for burn marks that indicate a control board failure and replace the control board if you find burn marks on the circuit board.
If your range hood uses a user interface control to start the fan, you may need to replace the user interface if it isn't sending signals to the electronic control board when you press the fan button. A user interface control is a keypad and display panel that sends signals to the electronic control board through a ribbon wire cable when you press its selection buttons. DIY testing of the user interface control isn't available for most range hoods. If the electronic control board doesn't activate the fan when you press the fan button on the user interface control, you'll likely need to replace the electronic control board. If the fan still doesn't work after replacing the electronic control board, you'll likely need to replace the user interface control.
If your range hood uses a switch to activate the fan motor, test the fan switch for continuity using a multimeter (with the range hood still unplugged) to see if the switch allows current to flow to the fan when you turn the switch on. Replace the fan switch if you measure no continuity through the switch with the switch turned on. If the switch or the electronic control board sends current to the fan motor but the fan motor doesn't run, replace the fan motor.