The impact of "nonlinearity" across the sciences is, well, nonlinear. If a chromotographer tells you your sample has a nonlinear calibration curve, tell him "boo hoo", rub your fingers together to indicate your playing the world's smallest violin and continue on. It simply means that he will have to work a little harder to put error bars on the analysis, rather than being able to apply a single % across the range.
But when the rheologist tells you that your sample is being tested in the nonlinear response range, run and don't ever come back. It isn't the rheologist that is complaining as nonlinear behavior in polymers is far more interesting.
To explain all of what is known of nonlinear rheology here would be impossible, so let me give you a small taste. For small enough deformations of your sample, the actual value of the deformation doesn't really matter much - you'll get the same output regardless of what the deformation is. All you really need to know is the stress and the strain rate. But if the deformation becomes too large, then everything - the strain, the strain history, the strain rate, and the strain rate history - and they are all interrelated by tensors whose manipulation becomes extremely complex if you're not using Cartesian coordinates.
But as I said, it's the nonlinear behavior that gives rheology all it's magic, such as the normal forces that we see in die swell and rod climbing.