You may well have a CCLI licence which permits you to project or print song lyrics in church. You are also permitted to play commercial CDs in church as part of a religious service. Those permissions are no longer valid when you stream your service on the internet. In short you can stream recordings of your own musicians playing, provided you have the correct licences, but you cannot use CDs, DVDs, YouTube videos or any other recordings of other performers without permission.
Websites and streaming
When a copyright work is used on a website or streaming service, it immediately gains worldwide availability, so there are strict limits on what you can do and most of the licences churches normally hold do not cover that context.
If a service or similar event is recorded and uploaded to the internet then you must have a streaming licence in order to show song lyrics. You can record your own versions of these songs (cover versions) but you cannot include other peoples’ recordings, e.g. ripped from a CD or another recording, without the express permission of the recording artist. A lot of people do it, but it is illegal and such recordings frequently receive take-down notices for copyright and performing rights violations.
The CCLI streaming licence covers the copyright but not the performing rights aspect. If you upload the recording to YouTube or Facebook then you are automatically covered for performing rights (royalty to the song/music author) of your own recording, as these organisations have an agreement with the Performing Rights Society which permits this.
YouTube has an automatic system for identifying copyright music, and on the whole it works pretty well, although occasionally it makes mistakes. If you upload your own recording of a copyright work you will very likely find a Copyright Claim notice on your video info. This is nothing to worry about, it just means that the copyright owner is entitled to claim any money you might have made from your video. If you upload a commercial recording you are likely to receive a Copyright Strike and that is much more serious and means YouTube thinks you have breached copyright. This may result in sound being muted on your video, or in you receiving a takedown request. It operates a three strikes rule, after which your account is likely to be suspended.
Zoom is also considered as a streaming service (so needs the streaming licence), but it does not include cover for performing rights. If you want to share your own performances you also need to purchase a Limited Online Music Licence from PRS to cover these performance royalties. This is an author royalty, not a performer royalty. And you cannot use commercial recordings without permission.
These same restrictions for Zoom would also apply if you upload content to your own website, so it is generally better to use YouTube and link to it from your website.
How to work around this
Fortunately there are several sources of music where the artist has given permission to re-use their work, which you can legally use in your own recordings. We’ve listed some of the ones we know about, both paid and free, on our Recorded Music Sources page.