I have only seen examples and references to the Navier-Stokes equations in two and three dimensions. Do we know if these equations work in one dimension or greater than three dimensions?
As a general matter, the Navier-Stokes equations work more or less the same way in any number of dimensions $d\geq 2$. Obviously, $d=3$ is the most important, but there are plenty of situations in which the flow may be well approximated by a $d=2$ solutions. The fact that there are only three spatial dimensions in the real world is the only particularl reason that $d\geq 4$ is rarely considered.
The situation with $d=1$ is different. With only one space dimension, there is no possibility for shear flow, which means that there is not that much interesting that can occur in the flow. For an incompressible fluid, in particular, there is only a uniform linear flow possible. (See, for example, this question from Math StackExchange: https://math.stackexchange.com/questions/1407491/which-of-these-1-d-representations-of-the-navier-stokes-equations-is-correct .)