We study subelliptic harmonic morphisms i.e. smooth maps ϕ:Ω→Ω~
among domains Ω⊂Rn and Ω~⊂RM endowed with Hörmander systems of vector fields X and Y, that pull back local solutions to HYv=0 into local solutions to HXu=0, where HX and HY 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 Ψ:M×(0,∞)→N×(0,∞) of the form Ψ(x,t)=(ϕ(x),h(t)) the map ϕ:M→N is a subelliptic harmonic morphism