  Let $X$ be a smooth projective toric variety and let $\widetilde{X}$ be the
blow-up manifold of $X$ at finitely many distinct tours invariants points of
$X$. In this paper, we give an explicit combinatorial formula of the Chow
weight of $\widetilde{X}$ in terms of the base toric manifold $X$ and the
symplectic cuts of the Delzant polytope. We then apply this blow-up formula to
the projective plane and see the difference of Chow stability between the toric
blow-up manifolds and the manifolds of blow-ups at general points. Finally, we
detect the blow-up formula of the Futaki-Ono invariant which is an obstruction
for asymptotic Chow semistability of a polarized toric manifold.
