I don't know of an example of a Riemannian manifold that admits a (natural) group structure, but not a Lie group structure.
The manifolds that I do know are various types of matrix manifolds (all of which are Lie groups), the circle, sphere, etc. (all of which admit $SO(n)$), and lorentz space-time (which does not have a natural group structure that I know of, and is hence excluded from the question).
I admit, the question is somewhat soft in nature due to the "natural" group structure condition.
Best Answer
The Riemannian metric is not really related to the question, so I will ignore it. As for the naturality it's not quite clear what you mean but I will always assume that you want topology and group structure to be compatible (in the sense that it is a topological group). Here are some references to possible interpretations of your question:
1) Given a (topological) manifold $X$, does there always exist a group structure that turns $X$ into a topological group? In general no - there are many obstructions like orientability, homogeneity, ... see the discussions here and here.
2) Given a topological group $X$ that happens to be a manifold, does $X$ always admit a differentiable structure that turns it into a Lie group? The answer is, even a unique one, see the discussion here.