A useful rule of thumb (though it is not always a sufficient condition for stability) is stability of the linearized scheme. Since you have a method of lines discretization, you can think of this geometrically as the condition that the eigenvalues of the jacobian of your spatial discretization, multiplied by the time step size, lie inside the region of absolute stability of your time discretization. In your case, this will give you the same condition that you would have for stability of an advection-diffusion problem, but with the advection speed given by the largest value of the initial data.
Of course, your backward difference in space will only be stable if the initial data is non-negative.
For sufficient conditions for stability of discretizations of nonlinear PDEs, see Strang's paper.
In the inviscid case, it's well known (and follows from the reasoning above) that this scheme is stable for CFL number $\le 1$.