Had lots of fun with this one. Although not the same type (Z77X-UD5H + i5-3570, BIOS F14), I might have a solution.
My symptoms:
- worked well without extra video card
- worked well with extra video card
- after unplugging the extra video card, the BIOS did not show a screen. Got 4 beeps (in morse: -...). System booted into OS, but no screen (my monitor did not get a signal).
With the extra video card I could enter the BIOS. I set "Init Display First" to "IGFX", but after reboot this flipped back to "Auto".
Tried to plug the extra video card in different sockets, but no change.
Tried to clear the CMOS, but it seemed that not enough was cleared: still no screen without extra video card.
As a last resort I reflashed the BIOS (same version, F14). After the verification was done, a message was shown about "Reset" (cannot remember exactly), and the system rebooted. I quickly flipped the power switch to off, removed the extra video card, turned the system on and got a screen!
So try reflashing your BIOS with the same version.
Regards, Nutz.