If $K$ is a field, it is easy to show using a basis that if $A$ and $B$ are $K$-algebras, then $Z(A\otimes_K B) = Z(A)\otimes_K Z(B)$ (where $Z$ denotes the center). This is no longer true if we replace $K$ by an arbitrary commutative ring $R$ (see https://mathoverflow.net/questions/137584/two-infinite-dimensional-algebras-such-that-the-center-of-their-tensor-product-i ; the first answer is actually incorrect, but I think Ben's example works).
But is it still true that a tensor product of central $R$-algebras is central? I am fairly convinced that it is not true, but I did not see any counter-example of the above result that uses central algebras; one reason is that it becomes true if at least one of the algebras is Azumaya, and Azumaya algebras are the most common source of examples of central algebras.
Nevertheless, does anyone know of a counter-example?