No - take $(\mathcal{P}(\mathbb{N}), \tau)$ where $\tau = \{\emptyset\}\cup\{A\subseteq\mathbb{N}: \mathbb{N}\setminus A \text{ is finite}\}$. Clearly, every two non-empty open sets have non-empty intersection, so the space is hyperconnected, but is not path-connected, see [this post][1].


  [1]: https://mathoverflow.net/questions/48970/why-are-the-integers-with-the-cofinite-topology-not-path-connected