Would an arduino and relay board kill a car battery?

liudr:
An arduino UNO draws 50mA at 5V so it will likely draw about 50mA at 12V.

If you use a regulator, yes. This would be a good time to spend the $2 on a buck converter!

Arduino has low-power modes but I wonder why keep it awake?

If you can live with lower clock frequencies you can also reduce power consumption, run at 3.4V and connect directly to SD (same voltage) at 8 MHz or less.

Other option: get the Arduino its own laptop battery and charge that when the engine RPM > idle + 500.