 Solution

No. Take $A=M_2(K)$ and its diagonal subalgebra $B$. The algebra over a field $B$ is commutative, so $Z(B)=B$. But $e_{11}\in B$ does not commute with $e_{12}$:Hence $e_{11}\notin Z(A)$, although $e_{11}\in Z(B)$. Commutation inside a subalgebra places no condition on elements outside it.