By definition of a $C^\ast$ subalgebra it is a closed subalgebra.
Why does it need to be closed?
This is a restriction that is not required in the case of a Banach subalgebra. (although I can't think of an example of an open Banach subalgebra example off the top of my head)
It needs to be closed because it needs to be complete. Completeness is part of the definition of a Banach algebra.
(Incidentally, notice that a set that is not closed need not be open. If a linear subspace of a normed space is open, then it must be the whole space.)