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 $10 = \frac{1}{2} { 6 \choose 3 }$ embedded tori.
See the remark on page 4 of my paper with Etingof, Kamnitzer, and Rains.
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..... wait, I need to check this.