FullStory playback should capture user interactions with videos that are a part of your site, with a few exceptions. FullStory does not capture the video itself.
If you have an embedded video hosted by YouTube, Wistia, or similar services you may notice that user interaction is not captured and seen at time of playback. This is due to the content being housed in a cross-domain iframe. If you don't have control over the iframe contents in order to insert the FullStory script, you won't be able to watch user interactions with the video during your playback.
Note: FullStory session replay will never show the actual video playing regardless of whether or not the video is self-hosted or hosted by a third party.