Let $\mathcal{E}$ and $\mathcal{F}$ be two coherent sheaves on a polarized projective variety $(X,\mathcal{O}\_X(1))$. 


Denote by $E=\Gamma\_\*(\mathcal{E})=\oplus\_{k\in\mathbb{Z}}\mathcal{E}(k)$, 
$F=\Gamma\_\*(\mathcal{F})$, and $R= \Gamma\_\ast(\mathcal{O}\_X)$.


Is it true that
$$\oplus\_{k\in\mathbb{Z}}Ext^i\_{\mathcal{O}\_X}(\mathcal{E}, \mathcal{F}(k))=Ext^i\_R(E, F)?$$ 


If the equality does not hold in general, when and to what extent it will hold? 



Ext modules of coherent sheaves and  associated modules
I assume you mean to ask whether:
$$\oplus\_{k\in\mathbb{Z}}Ext^i\_{\mathcal{O}\_X}(\mathcal{E}, \mathcal{F}(k))=Ext^i\_R(E, F)?$$ 
For this question, the answer is no, set $\mathcal{E} = \mathcal{O}\_X$. Choose $\mathcal{F}$ such that $H^i(X, \mathcal{F}) \neq 0$ for some $i > 0$. Then obviously 
$Ext^i\_R(R, F) = 0$. However $Ext^i\_{\mathcal{O}\_X}(\mathcal{E}, \mathcal{F}(0)) = H^i(X, \mathcal{F}) \neq 0$.


