  Let $n>1$ be an integer, $\alpha\in{\mathbb C}^n$, $b\in{\mathbb C}$, and $V$
a $\mathfrak{gl}_n$-module. We define a class of weight modules
$F^\alpha_{b}(V)$ over $\sl_{n+1}$ using the restriction of modules of tensor
fields over the Lie algebra of vector fields on $n$-dimensional torus. In this
paper we consider the case $n=2$ and prove the irreducibility of such
5-parameter $\mathfrak{sl}_{3}$-modules $F^\alpha_{b}(V)$ generically. All such
modules have infinite dimensional weight spaces and lie outside of the category
of Gelfand-Tsetlin modules. Hence, this construction yields new families of
irreducible $\mathfrak{sl}_{3}$-modules.
