Nothing at all. It's over 1,000 years old and in the public domain. If you include any music that is under copyright, you cannot monetize the video, but I'm unaware of any other restrictions. The being said, the YouTube robots will try to file copyright claims on chant recordings. It's annoying, but you can easily dispute them.
If you decide to reproduce a harmonization of the chant online that's fairly recent, you may need to tread lightly. Otherwise, it's long since entered PD.
Are you taking about a printed representation, or an actual sung instance?
Even if the tune and harmonisation are public domain, the actual performance is likely copyright, unless it was recorded <<however many>> years ago. Get yourself permission from the singers, or an appropriate licence for playing /streaming recorded music.
Apparently the new ICEL chants are copyrighted. I was part of a chant Webinar that involved singing part of the LOTH, but we couldn’t use the updated version because it was apparently under copyright by ICEL.
The printed representation. As far as I know in North America (and 100% in Canada), the physical layout of the page cannot be copyrighted, only the content. So even if it's a new origination of an older setting, it would still be fair game.
Perhaps this requires some clarification from the OP. I took "livestream" and "when a Gregorian chant is sung. i.e. the introit" to mean the broadcast of a live performance, not something prerecorded, not video of a printed edition, not ICEL chants in English.
Now here's a laugh. I just checked CMAA's collection of videos, and found that there was a YouTube "Content-ID" claim on one of them: https://youtu.be/YAbKXXb5pIQ?t=62 A demo of the Pater Noster, in Latin!
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