Lavrentieff proved a Theorem which implies that every real valued continuous function defined on a dense subset $D\subseteq \mathbb R$ admits a continuous extension to some $G_\delta $ subset of $\mathbb R$. See Theorem (4.3.20) in "General Topology" by Engelking, or this Mathematics Stack Exchange post.
If the dense subset $D$ is already a $G_\delta $, such as the set of irrational numbers, Lavrentieff's Theorem of course says nothing. So let us be a bit more audacious:
Given a real valued continuous function $f$ defined on $\mathbb R\setminus \mathbb Q$, is there an open subset $U$ of $\mathbb R$, containing all irrational numbers, and a continuous extension of $f$ to U?