Classroom - by Tom Woodward (CC) Creative commons BY-SA 2.0 (Modified) - https://www.flickr.com/photos/bionicteaching/14821910530
Mostly, - no. You need permission to play copyrighted material in the classroom, to students.
Section 28 of the copyright act 1968, for many organisations, allows for an exception to copyright for teaching purposes for those instructors playing copyrighted material in the classroom. However, unfortunately Section 28, however does not apply to MIT. This is because MIT is a for profit body.
This means MIT staff need permission of the copyright holder to play video and audio materials in a class setting, if the material is subject to copyright. Public domain material can of course be played in class, as can videos with the correct creative commons licence (eg creative commons on Vimeo).
If you have any specific questions about copyright at MIT, please email Bernard Lyons, Copyright Officer.