There are infinite dimensional weak Riemannian manifolds with vanishing geodesic distance (in the sense as defined in the question). These are modeled on nuclear Frechet spaces, but the results extend to Sobolev completions of high enough order ($>\dim/2 +2$). They are still weak Riemannian manifolds (i.e., the Riemann metric does not generate the topology on the tangent spaces).
The first example was the $L^2$ metric on $\text{Emb}(S^1,\mathbb R^2)/\text{Diff}(S^1)$, as shown in the first paper below. Then it turned out that the right invariant $L^2$-metric on each full diffeomorphism group also has this property, also Sobolev metrics for Sobolev order $<1/2$ ($\le 1/2$ on $\text{Diff}(S^1)$). In particular, Burgers' equation and KdV are nonlinear PDE's corresponding to geodesic equations for metrics with vanishing geodesic distance.
All the papers are in the arXiv or on my homepage.
Peter W. Michor; David Mumford: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS) 8 (2006), 1-48.
Peter W. Michor; David Mumford. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Math. 10 (2005), 217--245 (written later)
Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor: Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Glob. Anal. Geom. 41, 4 (2012) 461-472.
Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. Ann. Glob. Anal. Geom. 44, 1 (2013), 5-21.
Martin Bauer, Martins Bruveris, Peter W. Michor: Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II. 7 pages. To appear in: Ann. Glob. Anal. Geom.