How can we solve for $\int_{s} \Phi(\langle w, s\rangle) \mathcal{N}(s; \mu, \Sigma)\,ds$ i.e. expected value of probit over the inner product of Bivariate/Multivariate Gaussian Random Vector, where $\mathcal{\phi}$ is the probit function?
Essentially, I am trying to solve Expected value of applying the sigmoid function to a normal distribution but for the multi-variate case with the sigmoid over an inner product with some vector w.
I will use the notation $a^\top b$ instead of $\langle a, b\rangle$.
If $S \sim N(\mu, \Sigma)$ then $U := w^\top S \sim N(w^\top \mu, w^\top S w)$.
Your integral is $$E[\Phi(w^\top S)] = E[\Phi(U)].$$
Let $Z \sim N(0, 1)$ be independent of $U$, and note that $\Phi(u) = P(Z \le u) = P(Z \le U \mid U = u)$.
Then $$E[\Phi(U)] = E[P(Z \le U \mid U)] = P(Z \le U).$$
Finally, note that $Z-U \sim N(-w^\top \mu, w^\top S w + 1)$, so $$P(Z \le U) = P(Z-U \le 0) = \Phi\left(\frac{w^\top \mu}{\sqrt{w^\top S w + 1}}\right)$$
Note that the above argument is an adaptation of an answer that was mentioned in the comments of an answer to the question you linked.