If a space is hyperconnected (that is, the every non-empty open sets intersect), is it also path-connected?

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.