Showing a YouTube video to a public audience breaches copyright law, unless the specific video is released under a Creative Commons licence that explicitly permits public exhibition (which is very rare).
YouTube Terms of Service under 'Permissions and Restrictions', section 9, states you are not allowed to:
"... use the Service to view or listen to Content other than for personal, non-commercial use (for example, you may not publicly screen videos or stream music from the Service)"
