  Non-Commutative space-time introduces a fundamental length scale suggested by
approaches to quantum gravity. Here we report the analysis of the Casimir
effect for parallel plates separated by a distance of $L$ using a Lorentz
invariant scalar theory in a non-commutative space-time (DFR space-time), both
at zero and finite temperatures. This is done in two ways; one when the
additional space-dimensions introduced in DFR space-time are treated as extra
dimensions but on par with usual space-dimension and in the second way, the
additional dimensions are treated as compact dimensions. Casimir force obtained
in the first approach coincides with the result in the extra-dimensional
commutative space-time and this is varying as $\frac{1}{L^5}$. In the second
approach, we derive the corrections to the Casimir force, which is dependent on
the separation between the plate, $L$ and on the size of the extra compactified
dimension, $R$. Since correction terms are very small, keeping only the most
significant terms of these corrections, we show that for certain values of the
R, the corrections due to non-commutativity makes the force between the
parallel plates more attractive, and using this, we find lower bound on the
value of $R$. We show here that the requirement of the Casimir force and the
energy to be real, impose the condition that the weight function used in
defining the DFR action has to be a constant. At zero temperature, we find
correction terms due to non-commutativity, depend on $L$ and $R$ dependent
modified Bessel functions $K_{1}$ and $K_{2}$, with coefficients that vary as
$\frac{1}{LR^3}$ and $\frac{1}{L^2R^2}$, respectively . For finite temperature,
the Casimir force has correction terms that scale as $\frac{1}{L}$ and
$\frac{1}{L^3}$ in high-temperature limit and as $\frac{1}{L^2}$ and
$\frac{1}{L^4}$ in the low-temperature limit.
