I could not find a question where testing for white noise for non strick stationary non parametric time series is adressed. Per definition white noise is stationary. But finding a series that is stationary does not mean is white noise.
Traditionally testing for white noise is done via multiple tests for the three main characterictis of white noise. tests which adress the distribution question exist, however it is my intuition that tests under non-stationarity poses much more problems for results of variance, mean and autocorrelation.
For example in testing for independence (acf), constant variance, and zero mean, I am uncertain if the traditional tests for both parametric and non parametric times series adress the issue of non stationarity and it is my intuition provided that this tests are usually applied to ARMA, ARIMA all of which require non strick stationarity that this tests when applied to non stationary sources results may not be valid. (testing a time series which you dont know if is stationary or not and if is white noise or is not).
Outside the non traditional tests, white noise tests based on wavelets or spectrum of a series. The underlying methods are adequate for non stationary analysis of time series, therefore can it be said that this is also the case when testing for white noise as they use in general normalization. However I am uncertain on the adecuacy of these tests.
There is however a paper http://onlinelibrary.wiley.com/doi/10.1002/sta4.69/pdf where there is a comparison of the different methods but they are applied to ARIMA models.
In this respect it is my interest to find out what tests are adequate for determining if a time series is white noise under non parametric and non strick stationary sources.