There exist topological manifolds which don't admit a smooth structure in dimensions > 3, but I haven't seen much discussion on homotopy type. It seems much more reasonable that we can find a smooth manifold (of the same dimension) homotopy equivalent to a given topological manifold. Is this true, or is there a counterexample?
It is false for compact manifolds in 4 dimensions. Freedman showed that there is a compact simply connected topological 4-manifold with intersection form E8, but Donaldson showed that there is no such smooth manifold.
For every $n\ge 4$ there exists a closed aspherical topological $n$-manifold $N$ which is not homotopy-equivalent to a PL manifold. Furthermore, $\pi_1(N)$ is a CAT(0) group. This is a theorem of Davis and Januszkiewicz, see theorems 5a1, 5b1 in their "Hyperbolziation of polyhedra" paper http://intlpress.com/JDG/archive/1991/34-2-347.pdf The construction relies heavily on Freedman's result about E8-manifold. As Vitali said, this is an old post, but it is good to have an answer that works in all dimensions since dimension 4 is exceptional in many ways.