Can publishing on arXiv create copyright issues?
Nature especially states how pre-prints should be handled:


  Preprints
  
  Nature Research journals support posting of primary research manuscripts on community preprint servers such as arXiv and bioRxiv.  Preprint posting is not considered prior publication and will not jeopardize consideration at Nature Research journals.  [...] Authors posting preprints are asked to respect our policy on communications with the media (http://www.nature.com/authors/policies/embargo.html).
  
  [...]
  
  The original submitted version of the manuscript (the version that has not undergone peer review) may be posted at any time.  Authors should disclose details of preprint posting, including DOI, upon submission of the manuscript to a Nature Research journal. 


Taken from Nature's webside about Publishing licences and compliance with open access mandates.

TL;DR You can post your pre-print on arXiv prior to its submission to Nature.

Important: These condition may not only vary by publisher but by journal/conference.