Why are the white marbled leaves on my salix hakuro nishiki turning brown?

The white parts of the leaves have no chlorophyll (the green part) and so scorch easily in the sun. It is normal at this time of year for this to happen, especially in full sun. The tips could be lightly pruned back if it looks too unsightly but these browned leaves will probably drop soon anyway. The new growth next year will be fine, however the tips will most likely do the same thing. I wouldn't worry too much though. The shoots will have ripened to their tips by now so shouldn't die back over winter if left.
These same white bits are ironically what it is grown for, especially when they are bright pink earlier in the season.
It might be best to reconsider it's south facing position?

Keep it well watered even if it rains as it will need to get it's roots down to the moisture in the soil. It may be wind damage. You could temporarily protect it from wind with shading fabric stapled to stakes.