Probably the easiest way to tell if it needs a new battery would be to shut it down tonight and pull the power cord. If it boots up in the morning not knowing the time (make sure to pull the network cable too), then yeah, you will probably need to replace it. Another way to check will apply to only Tiger, and this might not be true. Anyhow, open System Information, and go under Power. See if it shows anything about the battery there. My Powerbook doesn't, since I know it doesn't have a watch type battery on the system board.

Also, to give you an idea of some bad errors in the system log, this is what my Powerbook was spouting a lot of a few months back:
Feb 20 00:51:50 dragonbook kernel: disk0s9: DMA failure.
Feb 20 03:35:51 dragonbook kernel: disk0: DMA failure.

Turned out to actually be the system board, as the problem persisted with a known good hard drive installed internally. Oddly, it took having the CPU set to highest peformance, and a high stress disk activity to reveal it. No diagnostics could find the problem, but installing World of Warcraft would reliably make the error appear in under a minute. That one had the local service people a bit baffled, since it passed their 24 hour stress testing fine. I'd bet by now they have a copy of WoW to test for problems :-)