The real locus of the moduli space of stable curves of genus zero with $6$ marked points is a non-formal space with vanishing Massey products.
This is a compact non-orientable 3-manifold that is hyperbolic on the complement of $6 \choose 3$ embedded tori.<br>
See the remark on page 4 of my [paper][1] with Etingof, Kamnitzer, and Rains.<hr>

The rational cohomology algebra of that manifold is generated by symbols
$\nu_{ijk}$ for $1\le i\lt j\lt k\le 5$, and has defining relations given by
$$
\nu_{ijk}\nu_{ijl}=0.
$$
See Proposition 2.3 of the above paper (the second relation in loc. cit. doesn't occur because $6$ is too small).

The Massey products vanish because their indeterminacy is too big: there is no room for them to be non-zero.

  [1]: http://arxiv.org/pdf/math/0507514v2