Does publishing a paper on arXiv prevent me from submitting it to a non-open access journal?
You are generally allowed to publish even in a non-open access journal even if a pre-print is on the arXiv.  Most journal copyright agreements explicitly allow the authors to post the article online.  Here's an example of a fairly generous one:


  The ASL hereby grants to the Author the non-exclusive right to reproduce the Article, to create
  derivative works based upon the Article, and to distribute and display the Article and any such derivative
  work by any means and in any media, provided the provisions of clause (3) below are met. The Author may
  sub-license any publisher or other third party to exercise those rights.


and a less generous one which still allows the author to post a copy online:


  I understand that I retain or am hereby granted (without the need to obtain further permission) rights to use certain versions of the Article for certain 
  scholarly purposes, as described and defined below (“Retained Rights”), and that no rights in patents, trademarks or other intellectual property rights 
  are transferred to the journal. 
  
  The Retained Rights include the right to use the Pre-print or  Accepted Authors Manuscript for Personal Use, Internal Institutional Use and for 
  Scholarly Posting; and the Published Journal Article for Personal Use and Internal Institutional Use.


I've seen examples where the journal actually did some genuine copyediting beyond what the referee did where the author wasn't allowed to post the version that benefited from the copyediting, but could still post the earlier version.

So, for most journals, the answer is that you're allowed to post the article online because it's specifically allowed by the document they ask you to sign.  But it is possible that posting on the arxiv will rule out particular journals that have more restrictive policies.