Let $X$ be a Noetherian scheme and $Z$ be a closed subscheme of $X$. Let $\pi: \tilde X\to X$ be the blowup along $Z$ so that $\pi|_{ \pi^{-1}(X\setminus Z)} : \pi^{-1}(X\setminus Z)\to X\setminus Z$ is an isomorphism (https://stacks.math.columbia.edu/tag/02OS) .
Then is it true that the closure of $ \pi^{-1}(X\setminus Z)$ is the whole $\tilde X$ ? If this is not true in general, then is it true if we put more hypothesis on $X$ (like $X$ is a variety or $X$ is affine) ?
We will need two lemmas.
Lemma 1: There is a 1-1 correspondence between effective Cartier divisors on a scheme $X$, and closed subschemes of $X$ whose ideal sheaves are invertible.
I will leave the proof of this as an exercise.
Lemma 2: The complement of an effective Cartier divisor in $X$ is open and dense in $X$.
Proof: Suppose $D$ is an effective Cartier, and let $U = X \setminus D$ be its complement. Clearly, $U$ is open. Showing the closure $\overline{U}$ is the whole of $X$ is a local question. So we may assume that $X = \mathrm{Spec} A$ is affine, and $D = V(f)$ for a non-zero divisor $f \in A$. Let $\overline{D(f)} = V(\mathfrak{a})$ for some (not necessarily unique) ideal $\mathfrak{a}$. Then for any prime $\mathfrak{p} \in D(f)$, $\mathfrak{a} \subset \mathfrak{p}$. Now since $f$ is a non-zero divisor, $D(f)$ contains all the minimal primes of $A$. Hence in particular, $\mathfrak{a} \subset \mathrm{Nil}(A)$, where $\mathrm{Nil}(A)$ is the nilradical. Thus $\overline{D(f)} = X$.
Now coming back to your question, if $\mathcal{I}$ is the ideal sheaf of $Z$, then the inverse image ideal sheaf $\pi^{-1}\mathcal{I} \cdot \mathcal{O}_{\tilde{X}}$ is an invertible ideal sheaf. Hence by Lemma 1 it corresponds to an effective Cartier divisor in $\tilde{X}$, whose complement is precisely $\pi^{-1}(X \setminus Z)$. Hence by Lemma 2 it is dense in $\tilde{X}$.