The monoidal category $(\mathbf{Ab},\otimes,R)$ has monomorphisms $f:M\to N$ and $f':M'\to N'$ such that $f\otimes f':M\otimes M'\to N\otimes N'$ is not monic, for example $f:\mathbb{Z}\to \mathbb{Z},k\mapsto 2k$ and $f'=\mathrm{id}:\mathbb{Z}/2\to \mathbb{Z}/2$.
I am wondering, if there is a similar example for epimorphisms. Precisely: Is there a monoidal category $(M,\otimes,1)$ such that:
$f_i:X_i\to Y_i$ are epic, but $f_1\otimes\dotsb\otimes f_r$ is not.
Yes, but for a silly reason: If $(M,\otimes,I)$ is a monoidal category, so is $(M^{\text{op}},\otimes,I)$. So your example involving monic arrows in $(\textbf{Ab},\otimes,\mathbb{Z})$ is also an example involving epic arrows in $(\mathbf{Ab}^{\text{op}},\otimes,\mathbb{Z})$.