As a modification to Alessandro Codenotti's comment, we can provide a ZFC example of a space (however, it is $T_1$ but non-Hausdorff) that admits non-trivial open covers (open covers $\mathscr U$ of $X$ so that $X \not\in \mathscr U$): consider $\mathbb R$ with its usual topology $\tau$ and a point $\infty \not\in \mathbb R$. Let $X = \mathbb R \cup \{ \infty \}$ have as its topology $$\left\{ U \subseteq X : U \in \tau \vee \left(\infty \in U \wedge X \setminus U \text{ is finite} \right) \right\}.$$ This space is clearly Rothberger and $\mathbb R$, as a subspace of $X$, inherits its usual topology; hence, $\mathbb R$ is a Lindelöf but non-Rothberger subspace of $X$.

If one desires the space to be non-compact, I believe the set $$\left\{ U \subseteq X : U \in \tau \vee \left( \infty \in U \wedge X \setminus U \text{ is countable, closed, and discrete} \right) \right\}$$ produces a non-compact topology on $X$ with the same property as above.