I got this message recently on a Tumblr blog I was setting up. I was editing the default (minimalist) theme to add a link in the header when I got this message:
Uh Oh! We couldn’t save your custom theme. Looks like your custom theme references non-HTTPS URLS. Please try again using only HTTPS URLS
It was a strange bug as I was adding a HTTPS URL, and had changed nothing else.
However, it seems that the theme itself had non-HTTP URLS by default and I needed to change those as well. A quick find and replace from HTTP: to HTTPS: seemed to fix the issue without any repercussions. I couldn’t find anything online about this so hopefully the inter webs and search engines of this world will make it available for others to see.