The best answer is…it depends.
Most classic hymns are not covered by copyright, and you can do whatever you want with them.
Newer songs, the ones your church needs a CCLI license to display the power point words for, are still copyrighted. However, depending on how you define "fair use", a few of these songs can be included on a video if they relate to the Pastor’s sermon or the general message.
However, you can get a license to play any song on TV. IT’s a royalty system. It’s not expensive, and I think that it’s really cheap if you’re only broadcasting on a local public access station.