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.