Would an arduino and relay board kill a car battery?

Remember that 'standard' car batteries - like SLA's don't like deep discharge cycles.
So unless you manage your discharge / recharge profile - the overall battery life will probably be severely degraded.

Are any relays energised when the battery is not charging (you are asleep?)
Of course as suggested above - sleep mode strategies will help - but the relays (and their controlled loads) will be the biggest hogs.