Under what situation is $sum_{i} p_{i}S(rho_i)$ > 0

Mathematically when is $sum_i p_i S(rho_i) > 0$?

I am assuming that ${p_i}$ form a probability distribution (and that none of the $p_i = 0$) and each $rho_i$ is a normalised state.

As $p_i geq 0$ and $S(rho_i) geq 0$ we have $sum_i p_i S(rho_i) = 0 iff S(rho_i) = 0$ for all $i$. Then we can ask the question under what circumstances do we have that each $S(rho_i) = 0$. But a state $rho_i$ has $S(rho_i) = 0 iff rho_i$ is a pure state.

To see this let ${lambda_x}_x$ be the eigenvalues of $rho_i$. We can then compute $S(rho_i) = - sum_x lambda_x log lambda_x$. Now as $rho$ is a positive semi-definite matrix we have for each $x$, $lambda_x geq 0$ and as $rho$ is normalized we have $ sum_x lambda_x = 1$. Putting these two constraints together we must have that for each $x$, $0 leq lambda_x leq 1$. This means that for each term in the sum we have $- lambda_x log lambda_x geq 0$. So $-sum_x lambda_x log lambda_x = 0 iff -lambda_x log lambda_x = 0$ for all $x$. But $- lambda_x log lambda_x = 0 iff lambda_x in {0,1}$. Combining this with the fact that we need $sum_x lambda_x = 1$ we must have exactly one $x$ for which $lambda_x = 1$ and the rest must vanish. Finally, if $rho_i$ is a state with a single nonzero eigenvalue then it is a pure state. Hence $S(rho_i) = 0 implies rho_i $ is pure. The other direction follows readily.

tl;dr $sum_i p_i S(rho_i) > 0 iff exists i$ such that $p_i>0$ and $rho_i$ is not pure.

An example Let $rho_{AB} = |psi rangle langle psi |$ where $psi = tfrac{1}{sqrt{2}}(|00rangle + |11 rangle)$. Suppose we measure on the first system the POVM ${M, mathbb{I} - M}$ where $M = frac{mathbb{I} + gamma sigma_z}{2}$, $gamma in [0,1]$ is some parameter and $sigma_z$ is the Pauli z operator. The $gamma$ parameter is sometimes referred to as the strength/sharpness of the measurement. When $gamma = 1$ the measurement is projective and when $gamma = 0$ the measurement is trivial (doesn't interact with the system). Labelling the measurement outcomes $0,1$ respectively we get outcome $0$ with probability $$ p_0 = mathrm{Tr}[(M^{1/2} otimes mathbb{I}) rho_{AB} (M^{1/2} otimes mathbb{I})] $$ and $$ p_1 = mathrm{Tr}[((mathbb{I}-M)^{1/2} otimes mathbb{I}) rho_{AB} ((mathbb{I}-M)^{1/2} otimes mathbb{I})]. $$ Similarly the normailised state on system B after receiving outcome $0$ is $$ rho_{B}(0) = frac{mathrm{Tr}_A[(M^{1/2} otimes mathbb{I}) rho_{AB} (M^{1/2} otimes mathbb{I})]}{mathrm{Tr}[(M^{1/2} otimes mathbb{I}) rho_{AB} (M^{1/2} otimes mathbb{I})]} $$ and on outcome $1$, $$ rho_{B}(1) = frac{mathrm{Tr}_A[((mathbb{I}-M)^{1/2} otimes mathbb{I}) rho_{AB} ((mathbb{I}-M)^{1/2} otimes mathbb{I})]}{mathrm{Tr}[((mathbb{I}-M)^{1/2} otimes mathbb{I}) rho_{AB} ((mathbb{I}-M)^{1/2} otimes mathbb{I})]}. $$ We can represent the correlations between system $B$ and the outcome of our measurement on system $A$ by some cq-state $$ rho_{A'B} = p_0 |0ranglelangle 0 | otimes rho_B(0) + p_1 |1ranglelangle 1 | otimes rho_B(1). $$ The entropy of this state is $$ S(rho_{A'B}) = H({p_i}) + sum_i p_i S(rho_B(i)), $$ like in your question. Now for the particular state we picked we can calculate the eigenvalues of both $rho_0$ and $rho_1$ to be ${(1+gamma)/2, (1-gamma)/2}$. So we see our states are pure only when $gamma = 1$. Note that this is exactly when the measurement $M$ is projective and not just a POVM. In general (I think but you should check) for any pure two-qubit state, if we measure one qubit with a binary projective measurement then the resulting reduced states on of the other qubit will be pure. And if we measure with a two-outcome non-projective measurement then the resulting reduced states will be mixed.

This situation frequently arises in cryptography where we measure a system which may be entangled with an adversaries system. Then we try to estimate how much information they have about our measurement outcomes given their quantum system. However this estimation usually uses conditional entropies like $S(A'|B) = S(A'B) - S(B)$.