At least in the compact case, there's a topological obstruction. In a 2005 paper, Mostow proved that a compact manifold that admits a transitive Lie group action must have nonnegative Euler characteristic. Here's the reference:
MR2174096 (2007e:22015)
Mostow, G. D.
A structure theorem for homogeneous spaces.
Geom. Dedicata 114 (2005), 87--102.
Of course, even if X admits a transitive Lie group action, most Riemannian metrics on X will not be homogeneous. (You didn't say whether you wanted actions by isometries, but I assume that's what you're interested in, because otherwise the Riemannian structure on X is irrelevant.) In the 2D case, the only compact, connected, homogeneous Riemann surfaces are the sphere, $RP^2$, the torus (and maybe the Klein bottle?)*, all with constant-curvature metrics. In general, the group has to be compact, because the isometry group of a compact Riemannian manifold is itself compact.
*EDIT 3: An earlier paper by Mostow constructed a transitive group action on the Klein bottle, but I doubt that this action preserves a Riemannian metric. It's a complicated construction, so I haven't had a chance to work through it in detail, but here's the reference:
Mostow, G.D., The Extensibility of Local Lie Groups of Transformations and Groups on Surfaces. Ann. Math., Second Series, (52) No. 3 (1950), 606-636.
I don't know what's known in the noncompact case.
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