Does the Boussinesq approximation imply 'incompressible' flow? Does the Boussinesq approximation imply that compressible inertial effects in the flow can be neglected (i.e. density can otherwise be treated as constant)?
It seems that it must be the case and perhaps it is obvious, since use of the Boussinesq approximation implies that the main driving force in the flow is natural bouyancy, in which case velocity will generally be low and $M<<1$ (Mach number).
Are there any known flow situations that are driven by buoyancy, where compressible inertial effects cannot be neglected?
 A: It depends on your configuration but for most purposes the density can be treated as constant (and so $\delta \rho << 1$). The most famous instance of this being incorrect comes from buoyancy driven flow, as you mention. In such a scenario, the gravitational acceleration $\mathbf{g}$ will be much larger than the local flow acceleration. You'll find that $$\delta \rho \mathbf{g} \sim \mu \nabla^2 \mathbf{u}$$ and so you cannot neglect the buoyant force from your momentum equation. Qualitatively this should make sense: the main driving force of your flow is buoyancy and hence it would be very silly to neglect it from your governing equation. 
To answer your question more generally, note that $$\begin{equation}
\frac{\delta \rho}{\rho_{0}}=\frac{1}{\rho_{0}}\left(\frac{\partial \rho}{\partial T}\right)_{p} \delta T=-\alpha \delta T
\end{equation}$$ where $\alpha$ is the so-called coefficient of thermal expansion. For most liquids, $\alpha \sim 10^{-4} K^{-1}$. If this is a small parameter in your problem then you can safely neglect it. 
A: To attempt to answer my own question:
It would seem that if the Boussinesq approximation is being used, then there is indeed an assumption that compressible inertial effects are negligible. Because, otherwise, if compressible inertial effects cannot be neglected, then density would be have to be related to temperature and pressure anyway, via an equation of state, in which case the Boussinesq approximation would not be appropriate or needed.
Essentially, the Boussinesq approximation is used to model the effect of density variations due to temperature, in a situation where changes of density are otherwise negligible (i.e. the flow can be considered incompressible).
