Will the fires in Australia cause warming due to CO2 or cooling as the smoke blocks out the sun?

As far as the globe goes it is an insignificant (probably unmeasurable) event.

But as far as the local effects, the stuff going up has variable effects. Like a volcano, which shoots CO2 and various particulates up. Each having specific effects. When massive Krakatoa erupted it was credited by historians with a cooling effect as far away as Europe for several years.

The smoke does have a cooling effect.  A couple of years ago we had by far the hottest day ever at my house, it was 112 F by 10:40 am.  About that time, a fire started on the east side of my small town and the smoke blotted out the sun for most of the day, cooling it down (to about 109) most of the day.  Late in the afternoon, the wind shifted and the sky cleared and the temperature went up to 113.  It probably would have reached 115 or 116 if it weren't for the smoke.

For the fires themselves i would guess that the smoke would have the bigger effect on world climate.

The overall effect would probably be negligible either way. I don't know what the exact content of bushfire smoke is, but I imagine the number of particulates is pretty low, so it won't effectively block or scatter sunlight over a large area.

In the short term, it might block out the sun to a degree (like volcanic eruptions often do).  But in the long term, only a REAL simpleton would think "combating climate change can be helped by burning huge amounts of forests down and killing BILLIONS (yes BILLIONS) of animals by doing it."