Auto LED dimly lit when off
Cars often pass a little bit of current through "off" bulbs in order to determine if the (previously incandescent) bulb was working normally. 

Depending on the replacement LED bulb circuitry, this can either cause intermittent flashing (when the LED driver circuitry builds up enough charge to light up), or constantly light an LED dimly (if it's a simple resistor in series with the LED). Yours sounds like the latter.

Olin is right with the suggested workaround - adding a bleed resistor in parallel with your LED bulb. The parallel resistor can provide an alternate path for the small "off" state current, meaning the voltage across the LEDs never gets high enough to cause them to light up.







simulate this circuit &ndash; Schematic created using CircuitLab

Please note that all values here are placeholders. You'll need to do some experimentation to find a bleed resistor value that works. 10k is probably a good starting point.

In "on" mode, the parallel resistor will just waste a tiny bit of power - it make the bulb visibly dimmer.

The replacement LED bulb will almost certainly be wasting less power than the original incandescent when "off". You could confirm that by checking the voltage across a working incandescent and your new LED bulb, and calculating the current through the bulb check resistor with V=I*R. Lower voltage across the bulb means higher voltage across the bulb check resistor, which means higher current wasted.