Selected answers from the Dumb SEO Questions G+ community.
Michael Martinez: Removing the listing will not in any way affect rankings or referrals. Google Search Console is mostly just a reporting tool, although you can use it for analysis and to report some problems fixed. If you are seeing the data under the HTTPS listing then it`s probably already been removed from the HTTP version of the site anyway. I monitor multiple Websites with both HTTP and HTTPS entries in GSC and the data is only reported in one place. It`s useful to have multiple listings to tell Google where to canonicalize your SERP URLs; but if you`re not publishing both HTTP and HTTPS content then after they get the canonicalization all sorted out the HTTP report is useless. So the complicated answer to your question is, if you`re absolutely 100% sure that the site is no longer publishing anything via HTTP then you don`t need the HTTP report in GSC; but if the site COULD BE or IS publishing content via HTTP then you should probably add the listing. You don`t need to do anything special for verification. GSC uses the same verification method you already have in place for the other protocol.