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The YouTube API has NO functions for actually streaming or playing videos. It only has functions for getting INFORMATION about videos and users, and listing videos by various criteria (most popular, featured, playlist, etc.) The latter function are erroneously referred to as "video viewing" functions in the API documentation, but they simply return information (including the URL needed to play the video) on groups of videos.
So, no, using the API will not cost you significant bandwidth. (Obviously, there is SOME bandwidth needed to retrieve the results from YouTube.)
But it's unclear that you need to use the API in the first place.