Google Webmaster Tools has added an International Targeting Report to show hreflang errors. There's a section that reports "no return tags" errors with Originating URL & Alternate URL columns.
What do I need to do to fix these errors? Is there a way to "Mark as Fixed" once that's done like we can for Crawl Errors?
This happens when a page includes a hreflang link to an alternate language, but the linked page doesn't link back to it. This Official Google Webmaster Central Blog post explains that:
Annotations must be confirmed from the pages they are pointing to. If page A links to page B, page B must link back to page A, otherwise the annotations may not be interpreted correctly.
So if you have this error reported, you should edit the Alternate URL page code to ensure there's a hreflang link back to the matching Originating URL.
Whereas the answer by Andrew makes sense and is in line with the official response by Google, I see 3 types of errors in my website:
A URL containing an URL-encoded URL is linked back using the properly encoded URL. E.g. http://example.com%3Flang%3Dzh is linked back as http://example.com?lang=zh - there is not much I can do if someone is linking my site using a wrong URL encoding.
The canonical URL is linked back through "x-default", but it is for some reason still detected as "no return tags"
The originating URL is listed as linking to an incorrect alternate URL. This is URL is not found either in the html of the originating page, nor in the sitemap (I am not using http headers). The head of the originating page is pointing to the correct URLs.
All these, especially 2 and 3, seem to be errors in the Google Webmaster Tools.
http://example.com%3Flang%3Dzh is likely to break and not actually get to your site (because the TLD is seen as com%3Flang%3Dzh). You probably mean something like: http://example.com/%3Flang%3Dzh