No. YouTuBox uploads your video files straight from Dropbox to YouTube without using your local connection. It tries to use the fastest, most direct available path between Dropbox and YouTube. Thus, none of your precious internet badwidth is consumed, once your video has been uploaded to your Dropbox account.
This makes YouTuBox a valuable tool in situations where your internet connection is slow or unreliable. Dropbox makes sure to reliably upload your files in the background. YouTuBox then moves your videos over to YouTube as fast as possible.