Note: This issue is now resolved in the Open edX Cypress release.
At the end of July, the YouTube Data API v2 will be deprecated. At that time, YouTube videos in courses on your instance of Open edX will fail to play.
EdX is working on a fix, which hosting providers will need to add to existing Open edX instances. This fix will also be included in the upcoming Open edX Cypress release.
For more information and to keep up to date on this issue, see the discussion in the edx-code Google group.