Your CMOS BIOS battery is supplying power in order to maintain the data stored in your bios. Having it depleted out on you when your PC is plugged probably won't do anything. However, if you unplug your computer with a depleted bios battery, your bios settings will reset back to its default settings. Your bios has two sources of power, the bios battery and the power from the wall. If you take out both of those, the bios will default. One source has to supply power to the CMOS in order to maintain its state. So your time stamp, boot sequence and others will probably change. The key is the boot sequence, if it resets it will probably try to boot form a disk drive, so your OS platform won't launch as it's stack pointed from your hard drive.
If you haven't already, take down the profile settings of your operational BIOS before changing out the battery. If your bios resets, then you have your settings down in paper. If you want to go hardcore, reflash your bios to where the default settings are your operational settings so in case the bios resets, it resets to an operational state.
No harm no foul should happen to your hardware if the bios battery goes out on you - just a matter of software settings and like you said the time stamp.