The software* is stored on the hard drive, yes. As for the version -- it will automatically be upgraded the next time the TiVo contacts tivo.com.
* Definitionally, it's not "firmware" unless it's in a ROM.
The TiVo does have a boot ROM, of course, similar to a BIOS. It checks the kernel to make sure it was signed by TiVo, and starts it, but that's about all.