An LED only requires 20mA of current to illuminate at a reasonable light level so I guess if there is leakage in the driver circuit to the LED then it is possible they will illuminate very dimly if the driver circuit does not have enough proper load drain to switch off.
Putting a shunt resistor in parallel to the LED's should suck any residual current out of the driver circuit and probably make the circuit switch off properly. A simpler "test" method would be to solder some wires to a working number plate bulb and attach this to the license plate terminals (hide it around the back somewhere taped up) - this will fool the driver circuit in working properly and your LED's should go out when the lights switched off. One bulb should be sufficient. If this works OK we can then get a power resistor to take the place of the bulb for a more permanent solution.