Do we know of any closed symplectic manifold $M$ with 2 cohomologous symplectic forms $\omega\_1$ and $\omega\_2$ such that there exist $\psi \in \text{Diff}(M)$ and $\psi^\* \omega\_1 = \omega\_2$ but $\omega\_1$ and $\omega\_2$ are not isotopic?


Here the word isotopy is supposed to mean that the two forms a joined by a path of cohomologous symplectic forms. 



Diffeomorphic but not isotopic symplectic forms
This answer is an extension of my last comment, which in turn is just a reference to [this answer](https://mathoverflow.net/a/34249/35687) by MO user Petya, and the paper [Symplectic Topology and Capacities](http://www.math.stonybrook.edu/~dusa/princerev98.pdf) by McDuff cited therein.


First, your condition for two forms to be isotopic is equivalent to the existence of a path of diffeomorphisms $\phi\_t$ such that $\phi\_{0} = \operatorname{id},\phi\_1^\*\omega\_2 = \omega\_1$. One direction is clear; in the other, let $\omega\_t$ be the family of symplectic forms and define a time-dependent vector field $X\_t$ by $\mathrm d\left(\iota\_{X\_t}\omega\_t\right) = \partial\_t\omega\_t$ (by assumption, the RHS is exact). The flow of $X\_t$, which exists by compactness, is the required family of diffeomorphisms.


Take $M = S^2\times S^2\times T^2$, with $T^2 = S^1\times S^1$ the torus, and let $\omega\_1 = p\_1\omega\_{S^2} + p\_2^\*\omega\_{S^2} + p\_3^\*\omega\_{T^2}$ where $\omega\_{S^2}$ and $\omega\_{T^2}$ are volume forms on $S^2$ and $T^2$, respectively. (Note that we use the same volume form for both factors of $S^2$.) Let $\psi(x,y,u,v) = (x,R(x,v)y,u,v)$, where $R:S^2\times S^1\to SO(3)$ sends $(x,v)$ to the rotation by $v$ around the axis through $x$. For fixed $y\_0,u\_0$, the map $x,v\mapsto (x,y\_0,u\_0,v)$ defines an $S^1$-family of $J$-holomorphic spheres for the product complex structure, and the projection of this family to the second $S^2$-factor si constant. Since each of these spheres has minimal positive area, there is no bubbling, so that for any deformation one obtains a compact cobordism between this family and its pushforward under the end of the deformation. In particular, the projection of the pushforward to the second factor is bordant to a constant map. But one can check that the map $f: S^2\times S^1\to S^2,(x,v)\mapsto R(x,v)y\_0$ is not bordant to a constant map: Its restriction to $S^2\vee S^1$ is nullhomotopic, and the resulting map $S^2\times S^1/(S^2\vee S^1)\cong S^3\to S^2$ is homotopic to the Hopf map, which is not nullbordant.


