We study subelliptic harmonic morphisms i.e. smooth
maps $\phi\colon \Omega \to \tilde{\Omega}$ among domains
$\Omega \subset \mathbb{R}^{N}$ and $\tilde{\Omega} \subset
\mathbb{R}^{M}$, endowed with Hörmander
systems of vector fields $X$ and $Y$, that pull back
local solutions to $H_{Y} v = 0$ into local solutions to $H_{X}
u = 0$, where $H_{X}$ and $H_{Y}$ are Hörmander operators.
We show that any subelliptic harmonic morphism is an open
mapping. Using a subelliptic version of the Fuglede-Ishihara
theorem (due to E. Barletta, [5]) we show that given
a strictly pseudoconvex CR manifold $M$ and a Riemannian manifold
$N$ for any heat equation morphism $\Psi\colon M \times (0,
\infty) \to N \times (0, \infty)$ of the form $\Psi (x,t)
= (\phi (x), h(t))$ the map $\phi\colon M \to N$ is a subelliptic
harmonic morphism.