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.
There is an extensive Atlas of Harmonic Morphisms (see http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/atlas.html) that contains a useful bibliography.