A typical LED draws between 15-30mA (0.015 to 0.03 amps) of current, depending on the color and brightness. Over 9 hours, thats 270mAh. A car battery has a much higher capacity. However, over time, this steady drain can cause drainage.
It's hard to know why the map light would stay dimly lit all the time, as there can be several different scenarios depnding on how things are wired in the car. Sometimes, the map light ground is switched, and if the LED is placed at the wire of the switch/light junction, enough current will flow through the LED to light the stock dome light and LED, albeit dimly, even with the switch open. It's not messed up, it's just how car is wired and how the LED is added to the circuit. There is a way to hook up LEDs so everything works right, but I haven't played with it yet, nor looked at the way everything is wired to offer more advice.