1D Burger's equation is not meant to model a physical phenomenon. Rather, it is a simplification of homogeneous incompressible Navier-Stokes equations that preserves (some of) its mathematical structure: the non-linear convection term and the second order derivative of viscous forces.
It was initially intended as a useful simplification to try to understand the mathematical problems present in N-S equations, e.g. the limit of small viscosities (high Reynolds number). It is not anymore a tool for researchers studying existence and uniqueness of solutions of N-S equations, but is still useful to test numerical schemes designed for N-S.
Its multidimensional version also arises in some other physical contexts, see http://arxiv.org/abs/nlin/0012033