Topology is of fundamental importance even to systems in classical mechanics. The configuration space (or phase space) of a generic classical mechanical system is a manifold and manifolds are topological spaces with some extra structure (e.g. a smooth structure in the case of smooth manifolds).
At the very start of any classical mechanics problem, you need to specify topological information.
For example, when we consider a single particle moving in one-dimension, we could consider a particle constrained to move either on a compact manifold (like a circle), or a non-compact manifold (like the entire real line). In each of these cases, the global topology drastically changes the nature of the solutions. If, for example, the particle were otherwise free in each of these cases, then in the compact case (the circle), the particle will always return to the same point after some finite time, while in the non-compact case (the real line), this cannot happen.
Addendum. Beyond it's fundamental importance to mechanics as described above, topological properties of classical mechanical systems are important for proving high-powered theorems about dynamical systems. If you, for example, open Foundations of Mechanics by Abraham and Marsden (which is really more a math than physics), then you'll find a chapter on so-called "topological dynamics" where you'll find results like corollary 6.1.9;
Let $M$ be a compact, connected, two-dimensional manifold, $X\in\mathscr X(M)$ and $A$ a minimal set of $X$. Then either (i) A is a critical point, (ii) A is a closed orbit, (iii) $A=M$ and $M=S^1\times S^2$.
Notice that the statement of this corollary depends on the topological assumptions that $M$ is compact and connected. There are all sorts of theorems like this in dynamical systems. See the Poincare Recurrence Theorem as another example.
Update. (2014 - July - 10)
More interesting information and discussion in the following physics.SE post:
What kind of manifold can be the phase space of a Hamiltonian system?
See also the links in the comments to that post, especially to mathoverflow.
Update. (2014 - July - 16)
Even more interesting related information and discussion in the following post:
Is there a physical system whose phase space is the torus?