  Following the Euclidean results of Varopoulos and Pankka--Rajala, we provide
a necessary topological condition for a sub-Riemannian 3-manifold $M$ to admit
a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group
$\mathbb{H}$. As an application, we show that a link complement $S^3\backslash
L$ has a sub-Riemannian metric admitting such a mapping only if $L$ is empty,
the unknot or Hopf link. In the converse direction, if $L$ is empty, a specific
unknot or Hopf link, we construct a quasiregular mapping from $\mathbb{H}$ to
$S^3\backslash L$.
  The main result is obtained by translating a growth condition on $\pi_1(M)$
into the existence of a supersolution to the $4$-harmonic equation, and relies
on recent advances in the study of analysis and potential theory on metric
spaces.
