Let $X$ be a nonsingular variety over $\mathbb{C}$, and let $\pi : \tilde{X} \rightarrow X$ be a blowing up of some codimension 2 set. Let $\mathcal{F}$ be a coherent sheaf on $X$.
If $H^i(\tilde{X}, \mathcal{F}) = 0$, then is it true in general that $H^i(X, \mathcal{F}) = 0$? Or, when is it true?
Of course, it is true for $\mathcal{O}_X$ because $H^i(X, \mathcal{O}_X)$ is birational invariant.
I don't understand exactly what you are asking for (since $\mathscr{F}$ is a sheaf on $X$, not $\tilde{X}$), but here is a partial answer:
Proposition. Let $X$ be a nonsingular variety over $\mathbf{C}$, and let $f\colon Y \to X$ be a proper birational morphism. Then, for every locally free $\mathcal{O}_X$-module $\mathscr{E}$ of finite rank and every integer $i \ge 0$, we have $$H^i(X,\mathscr{E}) \simeq H^i(Y,f^*\mathscr{E}).$$
Proof. Recall the Leray spectral sequence [Stacks, Tag 01F2], which says that for an $\mathcal{O}_Y$-module $\mathscr{F}$, there is a spectral sequence $$E_2^{pq} = H^p(X,R^qf_*\mathscr{F}) \Rightarrow H^{p+q}(Y,\mathscr{F}).$$ If $\mathscr{E}$ is a locally free $\mathcal{O}_X$-module of finite rank, then setting $\mathscr{F} = f^*\mathscr{E}$ in the spectral sequence above, we have $$E_2^{pq} = H^p(X,\mathscr{E} \otimes R^qf_*\mathcal{O}_X) \Rightarrow H^{p+q}(Y,f^*\mathscr{E}).$$ But by a theorem of Hironaka [Hironaka, Ch. 0, §7, Cor. 2], we have that $R^qf_*\mathcal{O}_X = 0$ for all $q > 0$, hence the spectral sequence degenerates, and we have the desired isomorphism. $\blacksquare$
This statement is moreover true when $X$ and $Y$ are schemes which are only assumed to have pseudo-rational singularities by [Kovács, Cor. 9.19].