  We study rank $r$ cohomological Donaldson-Thomas theory on a toric Calabi-Yau
orbifold of $\mathbb{C}^4$ by a finite abelian subgroup $\mathsf\Gamma$ of
$\mathsf{SU}(4)$, from the perspective of instanton counting in cohomological
gauge theory on a noncommutative crepant resolution of the quotient
singularity. We describe the moduli space of noncommutative instantons on
$\mathbb{C}^4/\mathsf{\Gamma}$ and its generalized ADHM parametrization. Using
toric localization, we compute the orbifold instanton partition function as a
combinatorial series over $r$-vectors of $\mathsf\Gamma$-coloured solid
partitions. When the $\mathsf\Gamma$-action fixes an affine line in
$\mathbb{C}^4$, we exhibit the dimensional reduction to rank $r$
Donaldson-Thomas theory on the toric Kahler three-orbifold
$\mathbb{C}^3/\mathsf{\Gamma}$. Based on this reduction and explicit
calculations, we conjecture closed infinite product formulas, in terms of
generalized MacMahon functions, for the instanton partition functions on the
orbifolds $\mathbb{C}^2/\mathbb{Z}_n\times\mathbb{C}^2$ and
$\mathbb{C}^3/(\mathbb{Z}_2\times\mathbb{Z}_2)\times\mathbb{C}$, finding
perfect agreement with new mathematical results of Cao, Kool and Monavari.
