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Yes, for example if $K \subset L$ is an inclusion fields, then the induced map Spec $L \to$ Spec $K$ is a homeomorphism (both source and target are single points), but the induced map on sheaves is the given inclusion of $K$ into $L$, which is surjective only if $K = L$.

For another example, let $X'\to Y$ be a closed immersion of schemes over ${\bar{\mathbb F}}_p$, and let $X \to X'$ be the relative Frobenius morphism. Then $X\to X'$ is a homeomorphism on underlying topological spaces but is not an isomorphism of schemes, and so the composite $X\to Y$ is a closed embedding on underlying spaces but not a closed immersion of schemes.

As one last example, let $X'$ be the cuspidal cubic given by $y^2 = x^3$ in the affine plane $Y$ (over $\mathbb C$, say), and let $X$ be the normalization of $X'$ (which is just the affine line). Then $X \to X'$ is a homeomorphism on underlying spaces, but is not an isomorphism of schemes. The composite $X \to Y$ is thus not a closed immersion, but induces a closed embedding of underlying topological spaces.

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Yes, for example if $K \subset L$ is an inclusion fields, then the induced map Spec $L \to$ Spec $K$ is a homeomorphism (both source and target are single points), but the induced map on sheaves is the given inclusion of $K$ into $L$, which is surjective only if $K = L$.

For another example, let $X'\to Y$ be a closed immersion of schemes over ${\bar{\mathbb F}}_p$. Then $X\to X'$ is a homeomorphism on underlying topological spaces but is not an isomorphism of schemes, and so the composite $X\to Y$ is a closed embedding on underlying spaces but not a closed immersion of schemes.

As one last example, let $X'$ be the cuspidal cubic given by $y^2 = x^3$ in the affine plane $Y$ (over $\mathbb C$, say), and let $X$ be the normalization of $X'$ (which is just the affine line). Then $X \to X'$ is a homeomorphism on underlying spaces, but is not an isomorphism of schemes. The composite $X \to Y$ is thus not a closed immersion, but induces a closed embedding of underlying topological spaces.