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 4manifold 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 homotopyequivalent 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/1991/342347.pdf The construction relies heavily on Freedman's result about E8manifold. 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. 

