I installed a PCIe FW card. The PCIe bus strips are at the bottom of the case, so the power connections won't reach the bottom.
No problem, I'll just use the wall wart.
Then I started wondering if there will be any problems using a 6-pin cable with the power adapter, i.e. will I fry the computer or do I need a 4-pin cable?

I'm assuming you didn't connect a 4-pin molex power lead to the Firewire controller?
If that's the case, there won't be any issue.
The card just won't have enough power to "bus-power" peripherals.
 
FWIW, you really shouldn't be using bus-power (if the audio interface has a dedicated power adapter).
Using bus-power can compromise fidelity (low power to condenser mics).

Thanks, Jim. I remember you telling me to use the wall wart in the past, I was just worried about possibly electrocuting myself or the computer with a 6-pin cable.
No, the molexes won't reach the FW card.

Good advice on using the wall wart!. I had issues with latency and track counts on bus power too, with my Motu traveller mk3. When I was using it.