Why does Earth have abundant oxygen in the atmosphere? Because of photosynthesis, obviously.  But then it's not actually that obvious after all, because photosynthesis is mostly balanced by respiration. 

We can summarise the processes of photosynthesis and respiration like this:
$$\begin{align}
\ce{CO2 + H2O + h\nu} &amp; \rightarrow \ce{O2 + CH2O(biomass)} \\
\ce{O2 + CH2O(biomass)} &amp; \rightarrow \ce{CO2 + H2O}
\end{align}
$$
If these two reactions were always balanced then the concentration of $\ce{O2}$ would never have changed. If I understand correctly, there are abiotic processes that deplete $\ce{O2}$ to form iron oxides, and so in order for there to be persistent $\ce{O2}$ in the Earth's atmosphere, the biosphere as a whole must continually produce an excess of it. In order for this to happen, there must be some processes that consume biomass at a non-neglible rate without removing a corresponding amount of $\ce{O2}$ from the atmosphere.

My question is, what are the processes primarily responsible for this? Is it just due to the burial of unoxidised organic matter (to form coal, oil, etc.), or is there more to it than that?

If they are different processes, I'm interested both in what caused the excess of oxygen during the "great oxidation" event, and what maintains the excess of oxygen on the modern Earth.
This is only a partial answer as it doesn't explain why the excess of O2 stayed but one thing you have to appreciate is the fact that aerobic respiration appeared almost half a billion years after photosynthesis, so we can't really say that photosynthesis and respiration have always balanced each other.  

Cyanobacteria (and with them photosynthesis) are believed to have appeared somewhen between 3 and 2.5Ga (e. g. Altermann &amp; Kazmierczak 2003, Brocks et al. 2003) while Mitochondria (and with them aerobic respiration) are thought to have evolved somewhen between 2 and 1.5Ga (see e. g. Hedges et al. 2004).

There was therefore at least half a billion year during which cyanobacteria were producing O2 with no one able to breathe it.