Examples of compact complex manifolds for which the $dd^c$ lemma does not hold

A known consequence of the $dd^c$-lemma is the vanishing of Massey products, which are certain secondary cohomology operations, see Deligne, Griffiths, Morgan, Sullivan, Real homotopy theory of Kähler manifolds, Inventiones 1975. However, the Iwasawa manifold (cf Griffiths-Harris) is a compact complex manifold with nontrivial Massey products.

Gauduchon proved that a compact complex manifold satisfies the $dd^c$ lemma for $(1, 1)$-forms if and only if $b_1 = 2h^{0,1}$. As a compact complex surface is Kähler if and only if $b_1$ is even, a compact complex non-Kähler surface does not satisfy the $dd^c$ lemma.

The reference for the above result of Gauduchon is the paper

Gauduchon, P. La classe de Chern pluriharmonique d’un fibré en droites, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 9, Aii, A479–A482

a link to which can be found in the comments below. Better still, an elementary proof of this result is given in the exercises from a course taught by Valentino Tosatti. A link to the exercises, complete with solutions, also appears in the comments. Thanks to YangMills for both links.