Let $(M,\omega)$ be a closed symplectic manifold. Assume $M$ is equipped with a Hamiltonian $T^r$-action, $r \in \mathbb{N}$ and let $\mu:M \to (\mathbb{R}^r)^*$ be its moment map. Let $A = \mu^{-1}(y)$ be a regular level set of $\mu$, $i: A \hookrightarrow M$ be the inclusion. Consider the symplectic quotient $(M_{\text{red}},\omega_{\text{red}})$ with $M_{\text{red}} = A/T^r$ and with $\omega_{\text{red}} \in \Omega^2(M_{\text{red}})$ satisfying $\pi^*\omega_{\text{red}}=i^*\omega \in \Omega^2(A)$, where $\pi:A \to M_{\text{red}}$ is the orbit space projection.
Q: Are there reasonable conditions under which the cohomology class $[\pi^*\omega_{\text{red}}]=[i^*\omega] \in H^2(A;\mathbb{R})$ is non-vanishing?
I think it would be difficult to come up with some 'tangible' criterion for the non-exactness of $\pi^* \omega_{\mathrm{red}}$, as there are very natural actions for which $H^2(A; \mathbb{R}) = 0$ (see for instance the fourth bullet below).
If $M_{\mathrm{red}}$ is zero-dimensional (that is, if $r = \frac{1}{2}\mathrm{dim}(M)$), then the form $\pi^* \omega_{\mathrm{red}} = 0$ is exact. So let us assume from now on that $r < \frac{1}{2}\mathrm{dim}(M)$.
Since $y$ is assumed to be a regular value of the moment map, $A$ is the total space a (principal) $T^r$-bundle over the (here closed) manifold $M_{\mathrm{red}}$. If that bundle admits a global section $s : M_{\mathrm{red}} \to A$, then $\pi^* \omega_{\mathrm{red}}$ cannot be exact. Indeed, suppose on the contrary that it were exact, so that there would exist an antiderivative $\lambda \in \Omega^1(A; \mathbb{R})$ to $\omega_{\mathrm{red}}$ i.e. $\mathrm{d}\lambda = \pi^*\omega_{\mathrm{red}}$. Then $s^*\lambda$ is an antiderivative to $\omega_{\mathrm{red}}$, meaning that the nonzero dimensional closed manifold $M_{\mathrm{red}}$ admits an exact symplectic form. This is absurd by Stoke's theorem, so $\lambda$ does not exist. The existence of a section $s$ is rarely met, as it is equivalent to saying that the (principal) $T^r$-bundle is trivial.
More generally, since the $T^r$-action preserves the closed manifold $A$ and preserves $\omega$, it preserves the form $\iota^*\omega = \pi^*\omega_{\mathrm{red}}$, so the orbit of $\lambda$ under the induced $T^r$-action on $\Omega^1(A; \mathbb{R})$ consists only in antiderivatives to $\pi^*\omega_{\mathrm{red}}$. Since $T^r$ is compact, this orbit is also compact, hence the average $\bar{\lambda}$ of this orbit $T^r \cdot \lambda$ inside the vector space $\Omega^1(A; \mathbb{R})$ is well-defined. Since averaging is a linear procedure and since the exterior derivative is a linear operator, we deduce $\mathrm{d}\bar{\lambda} = \pi^*\omega_{\mathrm{red}}$. But now $\bar{\lambda}$ is $T^r$-invariant, so it would induce a $1$-form $\bar{\lambda}_{\mathrm{red}} \in \Omega^1(M_{\mathrm{red}}; \mathbb{R})$ such that $\pi^* \bar{\lambda}_{\mathrm{red}} = \bar{\lambda}$ if $\bar{\lambda}$ were to vanish on the isotropic distribution on $A$. If that were the case, then we would again conclude that $M_{\mathrm{red}}$ is nonzero dimensional, closed and admits an exact symplectic form, which is absurd. However $\bar{\lambda}$ does not need to vanish on the isotropic distribution: given $\xi$ in the Lie algebra of $T^r$, let $X = X_{\xi}$ be the corresponding induced infinitesimal action of $A$. By invariance, $0 = \mathcal{L}_X \bar{\lambda}$. By Cartan's formula and since $X$ is tangent on $A$ to the i, $\mathcal{L}_X \bar{\lambda} = \mathrm{d}(\bar{\lambda}(X)) + X \lrcorner \, \iota^*\omega = \mathrm{d}(\bar{\lambda}(X))$. Hence each function $\bar{\lambda}(X_{\xi})$ is constant on $A$. The association $\xi \to \bar{\lambda}(X_{\xi}) \in \mathbb{R}$ is linear, so the kernel of this map is at least $(r-1)$-dimensional, but we can't get in general that it is $r$-dimensional. Hence it is not a very 'tangible' criterion.
Here is a 'reasonable' situation where $\iota^*\omega$ is exact. Consider the following Hamiltonian $T^1$-action on $\mathbb{C}^{n+2}$: $t \cdot (z_1, \dots, z_{n+2}) = (tz_1, \dots, tz_{n+1}, z_{n+2})$. It induces a Hamiltonian $T^1$-action on $\mathbb{C}P^{n+1}$: $t \cdot [z_1 : \dots : z_{n+2}] = [tz_1 : \dots : tz_{n+1} : z_{n+2}]$ whose moment map (in homogeneous coordinates) reads $\mu([z_1 : \dots : z_{n+2}]) = - \frac{\sum_{j=1}^{n+1} |z_j|^2}{2 \sum_{j=1}^{n+2} |z_j|^2}$. The affine chart $z_{n+2} = 1$ is invariant under this action, so we obtain a Hamiltonian circle action, in fact 'the standard circle action', on $\mathbb{C}^{n+1}$ 'equipped with the Fubini-Study symplectic form'. Take $A$ the sphere of radius $1$ in $\mathbb{C}^{n+1} \subset \mathbb{C}P^{n+1}$, so we know that $M_{\mathrm{red}} = \mathbb{C}P^n$ and $\omega_{\mathrm{red}} = \omega_{FS}$. But the corresponding 2-form on $A$ is exact, being the restriction of a 2-form on $\mathbb{C}^{n+1}$.