Space weather certainly could. As for dark matter, it would depend on the strength of the effect. If they were able to detect it with current GPS, it would need to be a temporal anomaly at least 10^-18 seconds in length, as that is the highest precision any current atomic clock (and would likely be 2-3 orders of magnitude higher or so, since the precision has been increasing over the past years, and GPS satellites likely are using some of the couple-year-old methods due to development + launch cycles).
Meanwhile, the speed of light is approximately 3 * 10^8 m/s. As such, with current GPS satellites, you likely would have a lower bound of around 10^-8 m of perturbation caused by any sort of signal messing with the timing which could be distinguished from noise.
From this:
http://www.unr.edu/nevada-today/news/2014/finding-dark-matterIt states: "If the dark matter causes the clocks to go out of sync by more than a billionth of a second we should easily be able to detect such events."
Which is actually quite a large deviation by the above calculations (unsurprising, since the above lower bound doesn't account for any sort of precision anomalies originating from the other equipment in the GPS aside from the clocks). A temporal anomaly of 10^-9 seconds times the speed of light is a full 30 centimeters; a good deal greater than the 5mm talked about in the article.