If Youtube doesn't show the subtitles, it's likely that they're not hardsubs. If you do have a file that has hardsubs they can't be hidden or formatted differently as they become an integral part of the frames for the movie. Think of it as using paint and adding text to every single frame. After you change that frame you won't be able to remove the text again in a fashion that allows you to see the original picture. This is because you've replaced part of the original data that made up the frame and that can not be recovered.
This setup also means that any distortions (e. g. caused by recompression) will affect the subtitles. This is in part why some of the DVD subtitles look so horrible.
If you look at the official handbrake documentation for the subtitle options you'll find that you have to check whenever you want handbrake to "burn in"/"hard burn" the subtitles in order to make them hardsubs.
If you're using a media player of your choice you should be able to deactivate subtitles if they're part of the file. Also be aware that some players will read subtitles files that are in the same directory as the file you're watching.