First, since you are dealing with closed manifolds, a submersion $\pi:X\rightarrow Y$ (assuming that the base is connected) is surjective because it is simultaneously open and closed. Moreover, due to Ehresmann's Theorem, the submersion $\pi:X\rightarrow Y$ is a fiber bundle as it is proper.

Here is an approach for constructing non-examples: Let $Y$ be a compact manifold with $\pi_1(Y)$ finite. Denote its universal cover by $p:\tilde{Y}\rightarrow Y$. Pulling back the fiber bundle $\pi:X\rightarrow Y$ with the finite cover $p:\tilde{Y}\rightarrow Y$ results in a new fiber bundle $\tilde{X}\rightarrow\tilde{Y}$. It is not hard to see that the natural map $\tilde{X}\rightarrow X$ is also a finite cover. Hence a multi-valued section $s:\tilde{Y}\rightarrow X$ lifts to a section of the bundle $\tilde{X}\rightarrow\tilde{Y}$ because $\pi_1(\tilde{Y})$ is trivial. So we need to arrange for the pullback of the bundle $\pi:X\rightarrow Y$ with the universal covering map $p:\tilde{Y}\rightarrow Y$ to do not admit a section.

1) Similar to @ThomasRot's comment, you may consider sphere bundles on $Y$: Let $E\rightarrow Y$ be an oriented vector bundle of rank $r$ whose Euler class
$e(E)\in H^{r}(Y,\Bbb{R})$ is non-zero. Now take $X$ to be the sphere bundle $S(E)\rightarrow Y$. If it admits a multi-valued section, from the discussion above the same must be true for its pullback $S(\tilde{E})\rightarrow \tilde{Y}$ where $\tilde{E}\rightarrow\tilde{Y}$ is the pullback of the bundle $E\rightarrow Y$ with the finite (universal) cover $p:\tilde{Y}\rightarrow Y$. But the former bundle cannot admit a section since the Euler class of the vector bundle $\tilde{E}\rightarrow\tilde{Y}$, given by $p^*(e)$, is non-zero because $p^*:H^{r}(Y,\Bbb{R})\rightarrow H^{r}(\tilde{Y},\Bbb{R})$ is injective.

2) Another approach is to take $\pi:X\rightarrow Y$ to be a principal $G$-bundle. Recall that such bundles are trivial if they admit a section, and their isomorphism classes are in bijection with homotopy classes of maps from the base to the classifying space $BG$. Thus any map $Y\rightarrow BG$ with the property that the composition
$\tilde{Y}\rightarrow Y\rightarrow BG$ is not null-homotopic provides you with a principal $G$-bundle $\pi:X\rightarrow Y$ without any multi-valued section.