Electrical – Ceiling Fan Light Only Works when Fan is On

Absolutely do not leave the wires alone wrapped in electrical tape. Electrical tape does not have enough longevity to safely protect the exposed leads and can create a hazard in the future.

The blue wire exists to support the future addition of a light. The polite thing to do for your future self or future owner of your home would be to connect the blue and red wires. However, it is not necessary; if you do not want to, you can cap both leads with wire nuts.

Note that it is also legitimate to connect the black on the fan to the red from the ceiling, and the blue on the fan to the black from the ceiling. Assuming you have two switches, that will swap which switch controls the fan and which switch controls the light.

If the existing black-black connection doesn't work, you can try the black-red. If that does work, it would suggest that the switch for the fan has failed, although it could also indicate other wiring problems.

Someone took a dangerous and illegal shortcut to separating the fan and light switches. Rather than running a second hot, they're using the white as a hot and the ground conductor as the neutral. You should rectify the situation immediately by returning the white wire to service as a proper neutral. We can help further if you describe or post a photo of the switch wiring.

Your final comment about having no neutral in the house makes sense if you're confusing neutral and ground. Older homes commonly have ungrounded branch circuits. At the switch, you probably have blacks and whites, but no bare grounds. My guess is that the ceiling light box was added after the house was built using contemporary 14/2 with ground. Since the ground didn't have a function, it was converted to be used as a neutral for a split switch scenario.