Does the Bernstein von Mises theorem require that the true posterior is actually Gaussian? Reading up on the Bernstein von Mises Theorem, it says that in the infinite data limit, the posterior converges to a Multivariate Gaussian.
Just a sanity check which I cannot find anywhere...This is clearly only true when the true posterior of the parameters is actually a Gaussian, right? If the true posterior is some other distribution that a Gaussian, this should not hold?
 A: Yes, but no.
The theorem gives conditions for the posterior to converge to a multivariate Gaussian.  A multivariate Gaussian is basically the only possible limit when the model has a fixed finite-dimensional parameter in the large-data limit and everything is at least a bit smooth, and that's one of the things the theorem shows. The other thing is that when you get convergence to a multivariate Gaussian it has the inverse Fisher information as the variance.
However, there are lots of reasonable settings where the theorem doesn't apply and the posterior doesn't converge to the sampling distribution of the MLE. One simple case is random-effects models where the number of parameters increases with the size of the data.  Another is spatial models where the number of parameters increases with the spatial scale of the data. Another is when the true parameter value is on the boundary of the parameter space. The basic form of the theorem also only applies when the observations are independent, though there are extensions to some other settings.
So, yes, it only applies when the large-data limit of the posterior is multivariate Gaussian, and that isn't always true, but it is true in an interesting special case of fixed-dimensional smooth models that the theorem addresses.
A: No, this actually holds quite generally! (as long as the conditions of the theorem are met).
As a simple example, consider the model
\begin{align*}
X_i &\stackrel{\text{iid}}{\sim} \text{Exp}(\lambda) \\
\lambda &\sim \text{Ga}(a, b)
\end{align*}
The posterior distribution of $\lambda$ is
$$\lambda|x_1, \ldots x_n \sim \text{Ga}\left(a+n, b + \sum_{i=1}^n x_i\right),$$
which is not "actually" a Gaussian posterior. In the limit however, as $n \rightarrow \infty$, the posterior will approach Gaussianity.
This holds in higher dimensions as well, and for a wide variety of problems.

A quick illustration. Lets look at the above posterior for different values of $n$ for data such that $\sum x_i = n/2$. Notice how, as $n$ increases, the posterior becomes more and more normally distributed.

