Consider a compact differentiable manifold $M$. We say that $f:M\to M$ and $g: M \to M$ are topologically conjugated if there exists $h:M\to M$ a homeomorphism such that $f\circ h= h \circ g$. The conjugacy class of a homeomorphism $f$ is the set of all $g$ such that $g$ is topologically conjugated to $f$.

If a homeomorphism $f: M\to M$ has infinite topological entropy (which is an invariant under topological conjugacy), then the conjugacy class of $f$ has no diffeomorphisms.

Is there any other known obstruction for a homeomorphism not to have diffeomorphisms in its conjugacy class? I would guess that yes, but I could not find one.

Is there a restriction on the dimension of the manifold?

(Remark: In dimension one, that is, in the circle, every homeomorphism has a diffeomorphism-of class $C^1$ in its conjugacy class. However, Denjoy counterexamples have no $C^2$ diffeomorphisms in theirs).