No doubt Noether's theorem holds for the symmetry of translations in space and time. But what if we zoom in on very small lengths and times, and spacetime maybe becomes discrete? Will this have consequences?
As Conifold pointed out in a comment, any discreteness of space–time is unlikely to take the form of a static lattice. (See the questions   that Qmechanic linked to for more on this.) But I'll anyway try to answer the question as posed: What if space is in fact a lattice with fixed spacing, so that the translation symmetries are actually discrete?
First note that a momentum $p$ is associated with a de Broglie wavelength $\lambda = h/p$. If there is a lattice spacing $a$ then only wavelengths longer than $a$ are meaningful, and so there is a maximum possible momentum $h/a$.
The consequence of the discrete translation symmetry is that momentum is still conserved, but only up to multiples of $h/a$. (We sometimes refer to it as quasimomentum.) A scattering event where momentum is conserved only modulo $h/a$ is called umklapp. This is in fact a very similar situation to that of phonons in a solid or the tight-binding model for electrons in a crystal lattice.
Does this have any consequences? Assuming the lattice spacing $a$ is of order the Planck length, then the maximum momentum $h/a$ is far too large to constrain any current experiments. And the energy required for an umklapp process to occur would also be of order the Planck scale.