For all practical purposes: no. The 12V accessory slot is fused at something more than 10A (I can draw 120W for a little inverter no problem) and 99% of dashcams are supplied with a 12V to 5V USB adapter that goes into the accessory slot to actually provide power to the camera. No camera on the market uses more than a couple watts, so you will never run into power issues with a dashcam. The car can support orders of magnitude more power.
That being said, if you intend on using the USB port (in the 1st gen Leaf) directly, that one is effectively limited at about 400-500mA. You will run into trouble with some more modern battery-backed dashcams that draw up to an amp when charging the battery. But again, that is easily fixed by using the accessory port and a USB adapter.
The accessory port is turned off when the car is off, so you run no risk of draining the battery when the car is off. Note that some cameras offer 'parking mode' or 'surveillance mode' recording, which means the dashcam keeps recording even if the car is off. This is done by using a battery inside the dashcam, not power from the car.