Is it true that Tensor product of injective modules is injective? Is it true that if $M$, $N$ are injective modules over a commutative ring $R$ (with identity) then $M\otimes_R N$ is also injective ?

It holds when $R$ is a PID (here injective $\Leftrightarrow$ divisible). It is not true in general. The following papers study and characterize this property of $R$ that tensor products of injectives are injective.

Ishikawa, Takeshi. "On injective modules and flat modules." Journal of the Mathematical Society of Japan 17.3 (1965): 291-296.
Enochs, Edgar E., and Overtoun MG Jenda. "Tensor and torsion products of injective modules." Journal of Pure and Applied Algebra 76.2 (1991): 143-149.

