Copyright law includes a concept called Fair Use. One of the four factors in Fair Use is the purpose and character of your use. Another is the effect of the use upon work’s value.
Playing a YouTube video in the classroom directly off the Internet is not a violation of Copyright or YouTube’s terms of service. Downloading the video so you can play it in the classroom with blocked Internet access is probably Fair Use and should not affect the work’s value. To ensure this, make sure you play the video all the way thru on your computer at least once. That way the creator will get credit for the video view and there will be no effect on the work’s value.
On the other hand, if you give copies of the video file to everyone in the classroom (lowering the work’s value), that would be crossing the line.
Giving everyone the URL link to the video (or embedding YouTube video links in a web page, as we do here) does not affect the work’s value as people are still viewing the video thru YouTube, and the creator will get the appropriate credit.
As with all things legal, your mileage may vary.
The best solution would be to just give teachers a way to bypass the firewall when needed.