Let $M$ be a closed complex manifold that is not deformation equivalent to a complex projective manifold.


Can $M$ be orientedly diffeomorphic to a complex projective manifold? What if $M$ is moreover Kähler?



Deformation equivalent vs diffeomorphic to projective manifold
I believe the answer is yes and follows from the combination of Theorem 4.6 here <https://arxiv.org/pdf/math/0111245.pdf>
and Theorem 1.3 here <https://arxiv.org/pdf/math/0111245.pdf>


The first result shows that deformations of standard complex tori are complex tori (i.e. $\mathbb C^n/\Gamma$ where $\Gamma\cong \mathbb Z^{2n}$). The second result shows that on $T^6$ there is an infinite dimensional family of complex structures.


**PS.** As for the second version of the question, where $M$ is asked additionally to be Kahler, I would guess that it can be safely counted as an open problem. Remember in <https://link.springer.com/article/10.1007/s00222-003-0352-1> Voisin solved negatively *Kodaira problem* by constructing the first ever example of a Kahler manifold that is not deformation equivalent to a projective one. It seems to me that since then no new examples were found of such a phenomenon. And as you pointed out in a different post, it was proven recently that in dimension $3$ every Kahler manifold can be deformed to a projective one, but this is hard work (strongly relying on dim $3$). So in order to advance in your question one has to advance in one of these two directions - trying to extend the $3$-dimensional result to dimension $4$ and trying to look for new constructions of Kahler manifolds...


