Gromov-Hausdorff convergence of compact metric spaces certainly do not preserve topology. However, at the level of fundamental group, for length spaces the following result is well known (I think):
Let $ \{ X_n \} _{n \in \mathbb{N} }$ be a sequence of compact length spaces converging to a compact length space $X$. Assume there is $ \varepsilon > 0 $ such that every ball of radius $ \varepsilon$ in $X$ is simply connected. Then for large enough $n$, there are surjective morphisms (for any choice of basepoints) $$\pi_1 (X_n) \to \pi_1(X)$$
When $X$ is a torus, the morphisms can be obtained from continuous maps $X_n \to X$.
Is there a reference for this result (or a result that implies this one)?