The answer is no, a result of Greene-Wu is that any connected, non-compact Riemannian manifold has a proper embedding by harmonic functions in some Euclidean space [see R. Greene and H.Wu: GREENE
Embedding of open riemannian manifolds by harmonic functions
Annales de l’institut Fourier, tome 25, no 1 (1975), p. 215-235, http://www.numdam.org/article/AIF_1975__25_1_215_0.pdf ].

However, any bounded harmonic function on a complete Riemannian manifold with constant volume is constant. In the book of Pigola and Setti [proposition 3.5 in Global divergence theorems in nonlinear PDEs and geometry. Ensaios Matematicos], you will more general result. But in the case of hyperbolic manifold with finite volume, I suspect that any harmonic function with exponential growth is constant.