Let $(C, \otimes, 1) $ be monoidal category with initial object $0$. Is it true that $0 \otimes a \simeq 0$ for all $a \in C$?
It is true in Set with cartesian product, $R-$modules with usual tensor product and $\mathrm{End}(A)$ the category of endofunctor of some category $A$ with composition. But is it true in general?
As your own answer shows, this is certainly true if the tensor product preserves finite coproducts (so, in particular, any closed monoidal category will satisfy this property).
However, to answer your original question, a simple counterexample would be to take any cocartesian monoidal category; that is, take the tensor product to be given by the coproduct (e.g., abelian groups with direct sum, or sets with disjoint union). In particular, the tensor unit is given by the initial object in this case! This means that if $\otimes$ is given by the coproduct, then $A\otimes0 = A$ for every object $A$.
Oh, I need to assume some compatibility of $\otimes$ with finite coproducts: $\left( \coprod_{i} a_i \right) \otimes a \simeq \coprod_i (a_i \otimes a)$. It is true in all those examples I gave. Now, $0 \otimes a = 0$ is just a tautology.
