  We prove the Pieri formulas for Schur multiple zeta functions, which are
generalizations of the Pieri formulas proved by Nakasuji and Takeda for hook
type Schur multiple zeta functions. Moreover, we also prove the
Littlewood-Richardson rule for Schur multiple zeta functions. In the course of
their proofs, we regard the `truncated' version of Schur multiple zeta
functions as series over $\mathrm{GL}(N)$ crystals to arrive at the
Littlewood-Richardson rule for the Schur multiple zeta functions.
