This question is a follow-up to this question which I asked on MSE.

Let $f: X \rightarrow Y$ be a surjective morphism of schemes, and $\mathscr{F}$ a coherent sheaf on $Y$. Are there conditions we can put on $X$ and $Y$ that imply that the global pullback map $H^0(Y, \mathscr{F}) \rightarrow H^0(X, f^* \mathscr{F})$ is injective?

If $f$ is flat or $\mathscr{F}$ is locally free, this map is always injective. In addition, this answer to the linked question on MSE indicates a proof that, at least when the schemes are noetherian and we either assume that $Y$ is regular or that $X$ and $Y$ are both normal, the same is true when $f$ is any finite map (of course, by miracle flatness, this is only interesting when $X$ is not Cohen-Macaulay or $Y$ is not regular). In the comments and edits to this question, we found counterexamples when $X \rightarrow Y$ is the normalization of a curve and when $X$ is smooth but not connected. Both of these are, to some degree, related to the phenomenon of $f$ "breaking apart infinitesimal neighborhoods". (Both of the modules considered were infinitesimal thickenings of points).

Can this sort of behavior occur when $X, Y$ are connected smooth varieties (ideally over $\mathbb{C}$)? It seems to me like the natural place to find a counterexample is when $f$ is a birational morphism and $\mathscr{F}$ has support on the image of the exceptional locus, but I have not been able to construct one along these lines.