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?

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.

