Here is my attempt at answering the question:
Proposition: Let $\mathcal{M}_0$ and $\mathcal{M}_1$ two nested models such that $\mathcal{M}_0 \preceq \mathcal{M}_1$. We note $\Theta_0$ and $\Theta_1$ the space of possible parameters for $\mathcal{M}_0$ and $\mathcal{M}_1$, with $\Theta_0 \subset \Theta_1$. If data generated from $\mathcal{M}_0$ and $\mathcal{M}_1$ are IID, then the following inequality holds $\forall \theta_0^* \in \Theta_0$:
\begin{equation} \label{eq:proposition1}
\langle \log p(\mathcal{D}|\mathcal{M}_0) \rangle _{p(\mathcal{D}| \theta_0^*,\mathcal{M}_0)} \geq \langle \log p(\mathcal{D}|\mathcal{M}_1) \rangle _{p(\mathcal{D}| \theta_0^*,\mathcal{M}_0)}
\end{equation}
If data are not IID, a sufficient condition for the inequality to hold is
\begin{equation} \label{eq:condition1}
k_{\mathcal{M}_0} \log (2 \pi) - \sum_{i=1}^{k_{\mathcal{M}_0}} \langle \log (\lambda_{i}^0) \rangle _{p(\mathcal{D}| \theta_0^*,\mathcal{M}_0)} \geq
k_{\mathcal{M}_1} \log (2 \pi) - \sum_{i=1}^{k_{\mathcal{M}_1}} \langle \log (\lambda_{i}^1) \rangle _{p(\mathcal{D}| \theta_0^*,\mathcal{M}_0)}
\end{equation}
where
$k_{\mathcal{M}_0}$ and $k_{\mathcal{M}_1}$ are the number of independent parameters of $\mathcal{M}_0$ and $\mathcal{M}_1$;
$H_0(\hat{\theta}_0)$ and $H_1(\hat{\theta}_1)$ are the Hessian matrices of the log-likelihoods $p(\mathcal{D}|\theta_0,\mathcal{M}_0)$ and $p(\mathcal{D}|\theta_1,\mathcal{M}_1)$ expressed at their respective MLEs;
$\{\lambda^0_i\}_{1 \leq i \leq k_{\mathcal{M}_0}}$ and $\{\lambda^1_i\}_{1 \leq i \leq k_{\mathcal{M}_1}}$ are the respective eigenvalues of $-H_0(\hat{\theta}_0)$ and $-H_1(\hat{\theta}_1)$.
Proof: using the same approximation as in the derivation of the BIC for $p(\mathcal{D}|\mathcal{M}_0)$ and $p(\mathcal{D}|\mathcal{M}_1)$ yields
\begin{gather}
\log p(\mathcal{D}|\mathcal{M}_0) = \log p(\mathcal{D}|\hat{\theta}_0,\mathcal{M}_0) +
\log \pi(\hat{\theta}_0|\mathcal{M}_0)+
\frac{k_{\mathcal{M}_0}}{2} \log (2 \pi) - \frac{1}{2} \log (|-H_0(\hat{\theta}_0)|)\\
\log p(\mathcal{D}|\mathcal{M}_1) = \log p(\mathcal{D}|\hat{\theta}_1,\mathcal{M}_1) +
\log \pi(\hat{\theta}_1|\mathcal{M}_1)+
\frac{k_{\mathcal{M}_1}}{2} \log (2 \pi) - \frac{1}{2} \log (|-H_1(\hat{\theta}_1)|)
\end{gather}
Both quantities then need to be averaged over $\langle \cdot \rangle_{p(\mathcal{D}| \theta_0^*,\mathcal{M}_0)}$. Assuming
\begin{equation}
\langle \log p(\mathcal{D}|\hat{\theta}_0, \mathcal{M}_0) \rangle _{p(\mathcal{D}| \theta_0^*,\mathcal{M}_0)} \approx \langle \log p(\mathcal{D}|{\theta}_0^*, \mathcal{M}_0) \rangle _{p(\mathcal{D}| \theta_0^*,\mathcal{M}_0)}
\end{equation}
(i.e. that the maximum likelihood estimator $\hat{\theta}_0$ will be close to the true value $\theta_0^*$ from which data were generated) yields $\langle \log p(\mathcal{D}|\hat{\theta}_0, \mathcal{M}_0) \rangle _{p(\mathcal{D}| \theta_0^*,\mathcal{M}_0)} \geq \langle \log p(\mathcal{D}|\hat{\theta}_1, \mathcal{M}_1) \rangle _{p(\mathcal{D}| \theta_0^*,\mathcal{M}_0)}$ (under Gibbs's inequality). Furthermore, $k_{\mathcal{M}_0} \leq k_{\mathcal{M}_1}$ yields $\pi(\hat{\theta}_0|\mathcal{M}_0) \geq \pi(\hat{\theta}_0|\mathcal{M}_1)$ (these quantities do not depend on $\mathcal{D}$). The inequality is thus met for the first two terms on the right-hand side.
For the last two terms, if data are IID and if the number of data points $T$ in $\mathcal{D}$ is sufficiently large, the same approximation as in the derivation of the BIC can be made:
$$
\frac{k_{\mathcal{M}}}{2} \log (2 \pi) - \frac{1}{2} \log (|-H(\hat{\theta})|) \approx -\frac{k_{\mathcal{M}}}{2} \log (T)
$$
Since $k_{\mathcal{M}_0} \leq k_{\mathcal{M}_1}$, the inequality thus holds if data generated from $\mathcal{M}_0$ and $\mathcal{M}_1$ are IID.
If data are correlated, the above approximation does not hold. However, the determinant of the Hessian (which is a symmetric matrix) can be written as the product of the eigenvalues, which finally leads to the necessary condition. This inequality can also be seen as a more general version of a result presented in the following paper using less stringent approximations :
Heavens, Alan F., T. D. Kitching, and L. Verde. "On model selection forecasting, dark energy and modified gravity." Monthly Notices of the Royal Astronomical Society 380.3 (2007): 1029-1035.
