What’s the small print… Do YouTube’s terms and conditions allow us to play video in Church?

In short, no. Here are part of the terms as of Feb 2017

They say:

1. you agree not to access Content or any reason other than your personal, non-commercial use solely as intended through and permitted by the normal functionality of the Service, and solely for Streaming.
3. You can read these terms yourself, they are nice and short! Pop over to YouTube and click the little Terms link at the very bottom of the page.
5. So, as a church service is (usually) a public service and definitely not a personal activity then the YouTube license does not allow playing of videos in church services.

2019 Update!

YouTube terms were updated in Summer 2019, the section on Permissions now says:

You are not allowed to:
9. use the Service to view or listen to Content other than for personal, non-commercial use (for example, you may not publicly screen videos or stream music from the Service);

Here’s a screenshot of the section:

So, YouTube still only allow playing of videos for your own personal use.