Here's a counterexample: over a field $k$, let $X=\mathbb{A}^4=\mathbb{A}^2\times \mathbb{A}^2$, and $U$ the complement of the origin. Put $Y=\mathbb{A}^2\times\{0\}$, $Z=\{0\}\times\mathbb{A}^2$, $X'=X\cap U$, $Y'=Y\cap U$.

Take $M=\mathcal{O}_{Y'}$ and $N=\mathcal{O}_{Z'}$. Then $M\otimes N$ is zero, while $ i_* M=\mathcal{O}_{Y}$ and $i_* N=\mathcal{O}_{Z}$, so $i_* M\otimes i_* N$ is the structure sheaf of the origin.

Here's a counterexample: over a field $k$, let $X=\mathbb{A}^4=\mathbb{A}^2\times \mathbb{A}^2$, and $U$ the complement of the origin. Put $Y=\mathbb{A}^2\times\{0\}$, $Z=\{0\}\times\mathbb{A}^2$, $Z'=Z\cap U$, $Y'=Y\cap U$.

Take $M=\mathcal{O}_{Y'}$ and $N=\mathcal{O}_{Z'}$. Then $M\otimes N$ is zero, while $ i_* M=\mathcal{O}_{Y}$ and $i_* N=\mathcal{O}_{Z}$, so $i_* M\otimes i_* N$ is the structure sheaf of the origin.

Here's a counterexample: over a field $k$, let $X=\mathbb{A}^4=\mathbb{A}^2\times \mathbb{A}^2$, and $U$ the complement of the origin. Put $Y=\mathbb{A}^2\times\{0\}$, $Z=\{0\}\times\mathbb{A}^2$, $X'=X\cap U$, $Y'=Y\cap U$.

Take $M=\mathcal{O}_{X'}$ and $N=\mathcal{O}_{Y'}$. Then $M\otimes N$ is zero, while $ i_* M=\mathcal{O}_{X}$ and $i_* N=\mathcal{O}_{Y}$, so $i_* M\otimes i_* N$ is the structure sheaf of the origin.

Here's a counterexample: over a field $k$, let $X=\mathbb{A}^4=\mathbb{A}^2\times \mathbb{A}^2$, and $U$ the complement of the origin. Put $Y=\mathbb{A}^2\times\{0\}$, $Z=\{0\}\times\mathbb{A}^2$, $X'=X\cap U$, $Y'=Y\cap U$.

Take $M=\mathcal{O}_{Y'}$ and $N=\mathcal{O}_{Z'}$. Then $M\otimes N$ is zero, while $ i_* M=\mathcal{O}_{Y}$ and $i_* N=\mathcal{O}_{Z}$, so $i_* M\otimes i_* N$ is the structure sheaf of the origin.