Disclaimer: I am not a lawyer. If you are seriously concerned about this issue, you should consult one; your institution probably has intellectual property lawyers on staff.
There is a general principle that "you can't copyright facts". Wherever you get your data set, you probably can legally publish any analysis of that data, without requiring anyone's permission. However, you may not be able to legally reproduce the data itself.
Of course, by standard academic ethics, you must properly cite and attribute the source of the data. And if you can't guarantee that the data will remain accessible, it could affect the reproducibility of your results and hence the quality of your paper.