Actually, the correct notion of maps preserving harmonicity is that of 'harmonic morphisms':  A map $f:(M,g)\to (N,h)$ between Riemannian manifolds is a *harmonic morphism* if it pulls back $h$-harmonic functions on $N$ to $g$-harmonic functions on $M$.  There are many, many nontrivial examples, and there is a large literature on the subject.