Should I be concerned about multicollinearity in nonparametric statistics?
Yes. Nonparametric vs. parametric is an unrelated issue from multicollinearity. If you have a multi-variable model (i.e., multiple input variables), then you should potentially be concerned about high levels of collinearity amongst those input variables.
Most of what we think of as prototypical nonparametric analyses are not multi-variable models, though.