Dash cams can drain a battery, but won't necessarily. Some dash cams will draw on a vehicle's battery when plugged in, even when the vehicle is off – this is how a few special features, like time-lapse videos or motion-sensing, are able to work when a vehicle is turned off.
As a rule of thumb, if the cam is plugged into a vehicle's OBD-II port, it can operate 24/7 with a minimum of battery-draining, thanks to the port's low voltage protection. If wired to the vehicle's ignition circuit, the dash cam won't draw power at all.
In addition, most dash cams can be set to turn off when the vehicle is off, and automatically turn back on when the vehicle turns on. Check your dash cam's manual to determine how it draws its power.