Does $\Gamma_*$ commute with tensor product? 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?
 A: It is already false in degree 0 : there is no reason why the natural map $H^0(\mathcal{F}_1)\otimes H^0(\mathcal{F}_2)\rightarrow H^0(\mathcal{F}_1\otimes \mathcal{F}_2)$ should be an isomorphism. This is already false for $n=1$, $\mathcal{F}_1=\mathcal{F}_2=\mathcal{O}_{\Bbb{P}^1}(1)$ .
A: 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 $\mathbb{P}^n$.
