  The Casimir effect is one of the most remarkable consequences of the non-zero
vacuum energy predicted by quantum field theory. In this paper we use a local
approach to study the Lorentz violation effects of the minimal standard model
extension on the Casimir force between two parallel conducting plates in the
vacuum. Using a perturbative method similar to that used for obtaining the Born
series for the scattering amplitudes in quantum mechanics, we compute, at
leading order in the Lorentz-violating coefficients, the relevant Green's
function which satisfies given boundary conditions. The standard
point-splitting technique allow us to express the vacuum expectation value of
the stress-energy tensor in terms of the Green's function. We discuss its
structure in the region between the plates. We compute the renormalized vacuum
stress, which is obtained as the difference between the vacuum stress in the
presence of the plates and that of the vacuum. The Casimir force is evaluated
in an analytical fashion by two methods: by differentiating the renormalized
global energy density and by computing the normal-normal component of the
renormalized vacuum stress. We compute the local Casimir energy, which is found
to diverge as approaching the plates, and we demonstrate that it does not
contribute to the observable force.
