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If the map $f^* \colon H^0(Y,\mathcal{L}) \to H^0(X,f^*\mathcal{L})$ is surjective then there is a commutative diagram $$ \require{AMScd} \begin{CD} X @>>> \mathbb{P}(H^0(X,f^*\mathcal{L})^\vee) \\ @VfVV @VVV \\ Y @>>> \mathbb{P}(H^0(Y,\mathcal{L})^\vee), \end{CD} $$ where the right vertical arrow is an embedding and the bottom arrow is a priori rational. But since $f$ is proper and dominant, it is surjective, hence the bottom arrow is regular. Moreover, it follows that $f$ is a closed embedding. But then $f$ is an isomorphism.