general rules of the variance of the posterior distribution In Bayesian posterior inference, we have the following equation relating posterior to prior and likelihood:
$$\pi(\theta|D)\propto \pi(\theta)l(D|\theta).$$
Is there any general rules that quantify the variance of the posterior vs the variance of the prior and variance of likelihood? Of course, without knowing the specific forms of the prior and likelihood, it is impossible to have any exact relations. But I am wondering if some approximate heuristics could be derived, potentially with some assumptions. In particular, if $\pi(\theta)$ is Gaussian and $D$ is also Gaussian around $\theta$, then $\pi(\theta|D)$ is also Gaussian and we can work out its variance. But is such an assumption reasonable? Or are there better estimates?
Having some estimate could be helpful, for example, to determine the approximate number of data points to acquire if a specific uncertainty level of the posterior is desired.
 A: 
Having some estimate could be helpful, for example, to determine the approximate number of data points to acquire if a specific uncertainty level of the posterior is desired.

This is actually pretty related to the field of "active learning". One key theorem that bounds a "variance" like term is given by Chernoff-Hofdinger bound which states in a simplified form that if $X_1, \dots, X_n$ are iid. random variables then
$$ P(|\overline{X} - E[\overline{X}]| \geq t) \leq 2\exp\left(-2nt^2 \right)$$
So the probability that your empirically estimated mean deviates from the true mean drops exponentially with the number of datapoints $n$ you consider. If you now want to compute the number of data points such that the probability $\delta$ of deviating from $\overline{X}$ at most by $t$. Then you can compute a lower bound by
$$ \delta \leq 2\exp\left(-2nt^2 \right) \iff n \leq \frac{\log(\frac{\delta}{2})}{-2t^2}$$
This would hold for arbitrary distributions; however; thus the bound is often too loose.

In particular, if π(θ) is Gaussian and D is also Gaussian around θ, then π(θ|D) is also Gaussian and we can work out its variance.

In this case, you consider a jointly Gaussian model of the data. This will perfectly work if your data is Gaussian-like distributed, much better than some general bounds. However if the data is not Gaussian-like your generative model is wrong, thus the posterior will most likely not concentrate around the true value of your parameter.
This approach is also much more general. If $\pi(\theta)$ is the conjugate prior to the likelihood $l(D|\theta)$ and if this prior has a closed-form expression for the variance, then you can also compute the posterior variance in closed form.
