About the only positive result that comes to mind is the fact that homomorphisms cannot increase the spectral radius, so that if the range space is a uniform algebra then homomorphisms are necessarily contractive.

In my view and in my experience, at the level of generality considered by this question, the assertion/hope that "a positive answer to the above question holds under very general conditions" is not backed up by evidence. The fact that the answer is negative even for bijective homomorphisms ${\bf M}_2 \to {\bf M_2}$, with both sides carrying the natural ${\rm C}^*$-norm, is one reason to doubt that much can be said in the noncommutative setting, without extra restrictions on the nature of the homomorphism.