A failed electronic control board or bad user interface control can prevent the lights and fan from working on a range hood. If the range hood uses switches to control the lights and fan, failed switches can prevent the fan from running and the lights from working. A wiring failure can also prevent the lights and fan from activating. Although rare, light socket failures and fan motor failure can occur at the same time.
Try replacing the light bulbs before doing any other tests. If the lights don't work after replacing the bulbs, unplug the range hood and check the wire harness connections. Reconnect any loose wires and repair any broken wires. Repairing a wiring problem may fix the lights and fan if the broken or disconnected wire supplied current to the lights and the fan motor.
If the wiring is okay, check the electronic control board and replace the control board if you see burn marks.
If your range hood uses a user interface control to start the fan and control the lights, you may need to replace the user interface if it isn't sending signals to the electronic control board when you press user interface buttons. 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 hood models so it's often difficult to tell whether the electronic control board or the user interface control is at fault when the lights and fan aren't getting power. If the electronic control board doesn't activate the lights and fan, you'll likely need to replace the electronic control board. If the lights and fan still don't work after replacing the control board, you'll likely need to replace the user interface control.
If your range hood uses switches to control the lights and fan, unplug the range hood and test the switches for continuity using a multimeter to see if the switches allow current to flow to the lights and fan when you turn the switches on. Measure resistance through the switches with the switches turned on. Replace the switches if the meter measures OL or infinite resistance indicates the switches won't allow current to flow to the fan and lights.
With new light bulbs installed and the range hood unplugged, check for continuity through the light sockets with new light bulbs installed. Replace the sockets if you don't measure continuity through the sockets with new bulbs installed. With the range hood still unplugged, check for continuity through the fan motor and replace the fan motor if you don't measure continuity through the motor.