Given a coherent sheaf $\mathcal{F}$ we denote by $\Gamma\_\*(\mathcal{F})=\oplus H^0(\mathcal{F}(d))$. Suppose, $\mathcal{F}\_1$ and $\mathcal{F}\_2$ are two coherent sheaves on $\mathbb{P}^n$. Denote by $M\_1$ (resp. $M\_2$) the modules $\Gamma\_\*(\mathcal{F}\_1)$ (resp. $\Gamma\_\*(\mathcal{F}\_2)$). Is it true that $M\_1 \otimes M\_2 \cong \Gamma\_\*(\mathcal{F}\_1 \otimes \mathcal{F}\_2)$? If not is there any condition on $\mathcal{F}\_1, \mathcal{F}\_2$ under which this is true?



Does $\Gamma_*$ commute with tensor product?
No, this is not true. For simplicity, let's work relative to an affine base scheme $\text{Spec} R$.
Denote $\Gamma\_\*(\mathcal{O}\_{\mathbb{P}^n})$ by $S$. Then $S$ is (naturally) a graded $R$-algebra that is isomorphic to $R[x\_0,\dots,x\_n]$.  

Take $\mathcal{F}\_1$ to be $\mathcal{O}\_{\mathbb{P}^n}(1)$, and take $\mathcal{F}\_2$ to be $\mathcal{O}\_{\mathbb{P}^n}(-1)$. 


With your definition of $\Gamma\_\*$, $M\_1$ equals $\tau\_{\geq 0}S[+1]$, i.e., the truncation above zero of the graded module $S$, shifted in degree by $+1$. And $M\_2$ equals $S[-1]$, i.e., the graded module $S$, shifted in degree by $0$. The tensor product $\mathcal{F}\_1\otimes\_{\mathcal{O}\_{\mathbb{P}^n}}\mathcal{F}\_2$ is isomorphic to $\mathcal{O}\_{\mathbb{P}^n}$. Thus the graded $S$-module $M:=\Gamma\_\*(\mathcal{F}\_1\otimes\mathcal{F}\_2)$ is isomorphic to the graded module $S$. And the multiplication map $M\_1\otimes\_S M\_2 \to M$ is just $$\tau\_{\geq 0}(S[+1])\otimes\_S S[-1] \to S.$$


This multiplication map is injective, and the image is $\tau\_{\geq 1}(S)$, the truncation of $S$ in degrees $\geq 1$. In particular, the map is not surjective. 


**Edit.** The answer by abx also addresses this issue (and abx answered first). I just want to mention one other issue. You could try to "fix" the problem from this example by redefining $\Gamma\_\*(\mathcal{F})$ to be the direct sum of $\Gamma(\mathbb{P}^n,\mathcal{F}(d))$ where $d$ varies over **all** integers, not merely the positive integers. That would "fix" the counterexample above. However, that just introduces a new problem. 


Let $L\_1$ and $L\_2$ be two lines in $\mathbb{P}^2$ that intersect in a single closed point $p$, say $L\_i = Z(X\_i)$ and $p=[1,0,0]$, where $[X\_0,X\_1,X\_2]$ are homogeneous coordinates on $\mathbb{P}^2$. Take $\mathcal{F}\_i$ to be the pushforward to $\mathbb{P}^2$ of the structure sheaf $\mathcal{O}\_{L\_i}$ of the line $L\_i$. Thus $\mathcal{F}:=\mathcal{F}\_1\otimes\_{\mathcal{O}\_{\mathbb{P}^n}}\mathcal{F}\_2$ is isomorphic to the pushforward of the structure sheaf $\mathcal{O}\_p$ of $p$.


Now, $M\_i$ equals $S/\langle x\_i \rangle$ for $i=1,2$. Therefore $M\_1\otimes\_S M\_2$ equals $S/\langle x\_1,x\_2 \rangle \cong k[X\_0]$. However, with the new definition of $\Gamma\_\*$, $\Gamma\_\*(\mathcal{F})$ is not $k[X\_0]$, it is $k[X\_0,X\_0^{-1}]$. So I do not believe there is any way to "fix" the definition of $\Gamma\_\*$ to entirely avoid the issues about compatibility with tensor product.


**Second edit.** The OP asks for counterexamples where $\mathcal{F}\_1$ is the structure sheaf of a closed subscheme. Again, there are counterexamples. In $\mathbb{P}^2$ with homogeneous coordinates $[X\_0,X\_1,X\_2]$, let $\mathcal{F}\_1$ be the structure sheaf of the line $L=Z(X\_1)$, so that $M\_1$ equals $S/\langle X\_1 \rangle$. Let $\mathcal{F}\_2$ be the structure sheaf of the conic $C=Z(X\_0X\_2-X\_1^2)$, so that $M\_2$ equals $S/\langle X\_0X\_2-X\_1^2 \rangle$. Then $M\_1\otimes\_S M\_2$ is isomorphic to $$S/\langle X\_1,X\_0X\_2 \rangle \cong R[X\_0,X\_2]/\langle X\_0X\_2 \rangle \subsetneq R[X\_0]\oplus R[X\_2].$$ However, $\mathcal{F}$ is isomorphic to the structure sheaf of $Z(X\_1,X\_0X\_2-X\_1^2)$, i.e., the disjoint union of the two closed points $p\_0=[1,0,0]$ and $p\_2=[0,0,1]$. Thus, the module $M$ is either $$S/\langle X\_1,X\_2 \rangle \oplus S/\langle X\_0,X\_1 \rangle \cong R[X\_0]\oplus R[X\_2],$$
if one uses the original definition, or
$$ R[X\_0,X\_0^{-1}] \oplus R[X\_2,X\_2^{-1}],$$
if one uses the second definition. Either way, the multiplication map $M\_1\otimes\_S M\_2 \to M$ is not surjective. This is one reason that some authors choose to *define* the homogeneous coordinate ring of a closed subscheme $Z\subset \mathbb{P}^n$ to be $S/\Gamma\_\*(\mathcal{I}\_Z)$ rather than $\Gamma\_\*(\mathcal{O}\_Z)$. However, even though this "fixes" the problem for structure sheaves of closed subschemes of $\mathbb{P}^n$, 
it is unclear (and unlikely) that this "fix" can be extended to all coherent sheaves on $\mathbb{P}^n$.


