Could you upload the rom somewhere, or check it's content ? is it a vgabios ? a EFI firmware ? does it have the "last rom indicator" bit set (see
http://forums.techpowerup.com/showthread.php?t=50701) ? This would let us know whether the dump is incomplete.
By the way, is it was dumped using GPUz, it means someone has windows (bootcamp) installed, which in turns means, that ati winflash can be used instead of the freedos trick, which is also a lot easier.
And when asking people to dump stuff, don't forget to ask them to report the size of the rom chip too
winflash usage is here :
https://forums.macrumors.com/posts/5849184/
There is one last possibility : the missing part of the rom could simply be in the machine's firmware, instead of the video card's firmware. This was done for instance with the first versions of the X1900XT firmwares : it only included the EFI part, and the vgabios part was inside the machine's EFI firmware, and loaded from here when starting bootcamp. But the fact that an official Mac 4870 works on a 2006 Mac Pro makes this a bit unlikely.
Edit : You shouldn't trust GPUz for rom dumping. It always dump the first PCI OpRom only, (except for 4870, where it now dumps the full 128k). So, it will NOT dump a usable rom image if there is an 2nd OpRom with EFI firmware in the rom, and it will NOT dump a usable rom image if your card does not match the vendor id and device id of a stock 4870.