Let us continue this discussion in chat.

You may apply Chow's lemma and minimal resolution to reduce to the case when $X$ is a smooth projective surface, no need to get so involved with such things. On the other hand, your $f$ is not a zero divisor of $R/I$, for otherwise, $C$ is a component of $D$ (your assumption).

But restriction $D|_C$ is just pullback of line bundle. So it is always Cartier?

@abx Thanks for the precise example! Indeed, any projective variety of dimension $n$ has a finite surjective morphism to $\mathbb{P}^n$.

@NickL The original question has many other assumptions, I just wonder what happens if removing all the assumptions. I would appreciate you copy this comment to an answer, I guess you mean $X:=$ blowup of $\mathbb{P}^2$, right? It has finite morphism to $\mathbb{P}^2$.

@NickL If $\pi$ is isomorphic, then $\tau:X\to Y\cong X$ is a birational automorphism which has to be an automorphism.

No need to consider the embedded points, I think. Or I made any mistake?

@NickL Thanks, I see!

@Henri Thank you very much for the example! $D|_D\sim 0$ since $\Lambda^2 E$ is trivial.

If the question is too general, one may focus on the case when $X$ is a normal projective surface.