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.
FullStory offers minimal support for the <video> tag. Meaning, FullStory will capture that the <video> tag was added to the Document Object Model (DOM), but it doesn't offer additional functionality such as sync playback with the timeline, etc. If the video has the autoplay attribute, the video may appear to play during session playback.