Let $X$ be a projective variety over $\mathbb C$ of dimension $n$. Let $\tilde{X} \to X$ be a resolution of singularities. Suppose that $H^n(\tilde{X}, \mathcal O_{\tilde{X}}) = 0$. What can we say about $H^n(X, \mathcal O_X)$? When is it $0$?
First, let me point out that $H^i(\tilde{X}, O_{\tilde{X}}) \cong H^i(X, O_X)$ if $X$ has rational singularities for all $i > 0$. Indeed, if $X$ has rational singularities if and only if
It immediately follows from the Leray spectral sequence that $$H^i(\tilde{X}, O_{\tilde{X}}) \cong H^i(X, O_X)$$ for all $i \geq 0$. In fact, for any Cartier divisor $D$ on $X$, the same argument implies that $$ H^i(\tilde{X}, O_{\tilde{X}}(\pi^* D) ) = H^i(X, O_X(D)) $$ for any $i \geq 0$ since the projection formula can be applied in the cases of 1. and 2. above. Now, without rational singularities, you can run into trouble. For example, suppose that $X$ is a normal CohenMacaulay variety with an isolated singularity $x \in X$ that is not rational. Consider the exact triangle in the derived category: $$O_X \to R \pi_* O_{\tilde X} \to C \xrightarrow{+1}$$ Because $X$ is a normal CohenMacaulay, and has an isolated nonrational singularity, we know $C = M[n+1]$ is a nonzero module supported at $x \in X$ (shifted over by $n1$). See Lemma 3.3 in Rational, Log Canonical, Du Bois Singularities: On the Conjectures of Kollár and Steenbrink by Sándor Kovács. Then we have the following exact sequence by taking (hyper)cohomology $$ 0 \to H^{n1}(X, O_X) \to {{H}}^{n1}(\tilde{X}, O_{\tilde{X}}) \to {\mathbb{H}^{n1}}(X, C) \to H^n(X, O_X) \to H^n(\tilde{X}, O_{\tilde{X}}) \to 0 $$ where the two end points are zero since $C = M[n+1]$ an Artinian module with a shift. On the other hand, $\mathbb{H}^{n1}(X, C) = H^0(X, M) \neq 0$ for the same reason. Now, if $\tilde{X}$ is for example Fano and we are in characteristic zero, then $$H^i(\tilde{X}, O_{\tilde{X}}) = H^i(\tilde{X}, O_{\tilde{X}}(K_XK_X)) = 0$$ by Kodaira vanishing for $i > 0$. But then $H^n(X, O_X) \neq 0$ from the exact sequence. Beyond the Fano case, you might luck out of course, but I don't see any reason why it would hold in general. 


A very easy counterexample: let $X$ be a nodal cubic in $\mathbb{P}^3$. Then the resolution of singularities of $X$ is $\bar X = \mathbb{P}^1$, so $H^1(\bar X, \mathcal{O}_{\bar X}) = 0$. On the other hand, the exact sequence of sheaves on $\mathbb{P}^2$ $$ 0\to \mathcal{O}(3)\to \mathcal{O} \to \mathcal{O}_X \to 0 $$ shows that $H^1(X, \mathcal{O}_X)$ is onedimensional. 

