When I turn off my computer at the end of the day, should I also turn off the computer's battery backup?
Do you mean an UPS? It should not be necessary, but it won't hurt, and might save a couple pennies in electricity costs. But it might also mean you will not have a fully charged battery in the morning - and that might mean a reduced run time, should you need it. That said, do you turn your computer off completely? That is, do you unplug the computer or flip the master power switch (if your PSU has one)? The ATX Form Factor standard requires ATX power supplies to provide +5Vsb standby power when the computer is turned off. If you shut down the UPS too, then no standby power. If you don't need any "Wake on..." features, then no big deal.
Thinking further this morning - slowly, but further. Again, assuming this is an UPS, what else is plugged into it? I run all my network gear (modem, router, wireless access points (WAPs), switch, and PDA charger/hotsync cradle), as well as my monitors through my UPS. If I were to power off my UPS too, that would shut down my network, and will likely cause issues with DHCP IP assignments on my network, and maybe from my ISP that will see my modem is off line. This means I would have to reset my network every morning and that would be a PITA.