Let $X$ be smooth variety of dimension $\operatorname{dim} X = d$, let $\mathcal{F} \in \operatorname{Coh}(X)$ be a sheaf supported on a subvariety $Y \subset X$ of codimension $\operatorname{codim}(Y, X) = 1$, and let $D$ be a divisor. How do the intersection products $$ D \cdot Y \text{ and } D \cdot \mathcal{F}$$ relate? More generally, if $\mathcal{F}$ is supported in dimension $r$, we can consider $$ D_1 \dotsm D_r \cdot \operatorname{supp}(F) \text{ and } D_1 \dotsm D_r \cdot \mathcal{F}.$$ In Higher-dimensional algebraic geometry, Debarre writes
[...] we may also define an intersection number $$D_1 \dots D_r \cdot \mathcal{F}$$ [...] but this apparently greater generality is illusory.
Unfortunately, he does not expand on this.
Some context: Suppose $X$ is a surface, $\mathcal{F}$ is supported on a curve $C$, and $D$ is an ample divisor. Then I want to conclude $$D \cdot \mathcal{F} > 0.$$ I know by the Nakai-Moishezon criterion, that $$ D \cdot C > 0,$$ but I don't know how to relate this to $\mathcal{F}$.