Time and date are not stored in ROM. It's the PIC that holds it when off.
The PIC is a separate, small microcontroller that acts as real time clock and power controller. When you shut down your Firebee, the main processor (Coldfire) requests the PIC to cut mains power from the system, then the PIC goes into low power mode and (basically) only preserves time and NVRAM while waiting for you to hit the power button again. In low power mode, it can sustain from the internal battery up to four months even if disconnected from mains.
That said, there is a known problem with recent BaS_gcc that sometimes prevents FireTOS to cold boot for no apparent reason while EmuTOS comes up without problems. For now, I unfortunately was unable to find whats causing this.
If your mainly using FireTOS, I'd recommend (although it hurts) to flash original BaS instead of BaS_gcc. BaS_gcc has advantages when using EmuTOS (like "true" ST-RAM, faster networking in MiNT, bug fixes for MMU support and other things) but doesn't improve anything for FireTOS since the latter reinitializes all the settings that BaS_gcc does later on anyway.
Long story short: for FireTOS usage, better use original BaS for now until the problem is identified and fixed. Make sure you flash it from FireTOS, not EmuTOS.