One of my ceiling fans takes three small bulbs. For years I’ve used those small candle-shaped incandescent bulbs. Looking for more light I purchased and installed three LED bulbs. Turn them on and all is fine. This where it really gets weird. Turn switch off and 2 go out, but the third still glows a bit. If I unscrew it one of the others goes on. Working on a tip, I replaced one of the LED bulbs with an incandescent. Now they all turn completely off.
Anyone have any idea what’s going on?
Is it using one of those illuminated, glow-in-the-dark switches?
Often those contain a small LED or neon bulb within the switch itself, which is lit by allowing a small leakage current even when the switch is ‘off’ – enough to light up the switch handle, but not to light up the incandescent bulbe. But some new LED or CFL bulbs are so sensitive that they begin to light up on that low leakage current.
What is controlling the light fixture? Simple wall switch? Wall dimmer? Chain hanging from fan? Wireless remote?
I’m guessing you nailed it. 
W/regard to the switch, it’s a remote. No wall switch. Does a wireless remote allow for leakage?
PS: I understand how leakage could be an issue. However, two questions remain:
Why would only one LED glow, and
Why, when one of the three is replaced with an incandescent, neither of the LEDs glow?
Check if you bought non- dimable variety of LEDS - they usually have this problem. Just buy the dimable ones and your problem will most likely be solved (same thing happened to me)
It’s the first one in the circuit and uses all the available electricity itself.
Incandescent bulbs use a lot electricity to glow. That’s why you never saw them glowing. Like answer #1, it uses up all the available electricity.
And a remote also needs electricity to work. The receiver on the ceiling fan requires some electricity to be on at all time.
So it’s a remote?
I am guessing the switch inside the fixture is not a mechanical relay but a solid state relay or solid state switch. In either case the switching element (e.g. triac, pair of SCRs, pair of MOSFETS) is never really “off”; it is high resistance when your turn off the lights. The voltage across each bulb is determined by a voltage divider comprised of the resistance of the switching element and the load resistance (i.e. the resistance of the three bulbs in parallel).
If each bulb is incandescent, the load resistance is very low, and the voltage across each bulb will be very low. You will not see them glow.
If one bulb is incandescent and the other two are LED, the load resistance is very low (it is “swamped out” by the one incandescent), and the voltage across each bulb will be very low. You will not see them glow.
If each bulb is LED, the load resistance is not low. I am not sure what it would be. But at low voltage levels, LED lights may have thousands of ohms (again, not sure). If all three LED lights were identical then each would glow dimly the same amount. In reality, though, they’re not identical. Which means the most sensitive or efficient light would glow the brightest.
And again, the above is just a guess based on the information thus provided.
What is the remote controlling? The fixture or the LED bulbs directly?
The remote is a simple “universal” unit. It controls the fan, the direction of the fan, the speed of the fan, the “down” light and the “up” light (if it had one). No dimming capability.
LEDs run on a trickle of current, which means you can’t use them with most timers, dimmers, illuminated wall switches or remote control fixtures. Nearly all of those have a small trickle of sensor or leakage current that will keep all or some bulbs powered enough to light.
(What irks me is that even the relay-type X-10 control modules have this leakage… why, I am not entirely sure.)
I made our dining room fixture work by using 4 LED bulbs and one incandescent. The indy pulls just enough current to shut down the others. It means the light draws somewhat more power than if it were all LED, but OTOH I didn’t have to replace an expensive remote dimmer with a doubly expensive, non-remote, LED-compatible one.