Shooting from the hip here, but:
1) I don't think there are any privacy issues with recording something in a public place.
2) I don't think there are any copyright issues with recording a public display of anything(otherwise we could all be in jail). Copyright issues arise with rebroadcast or sale.
Someone will certainly correct me.
The article said he's using footage from these tapes in videos that he's making. Whether he's selling his videos isn't clear, but I don't think that's a decisive point.
Regarding the privacy question, I know that we have to sign a release when our children participate in events such as Scouts service projects, so that pictures can be used by the news or the Scouts for publicity if they wish. If we don't sign, they can't use our children's pictures.
I don't know if this is a comparable situation (maybe for underage performers?), but that's why I mentioned privacy as an issue.
But, as he said, he uses portions of them in his own videos; this seems to be a clear violation of copyright as well as a violation of the rights of any actors in the productions who had not given their releases for the use of their images. I think this would hold even if he distributes the videos free for evangelical purposes rather than selling them. He may not have been sued as yet because the LDS church was unaware of his activities, or because they consider him a trivial nuisance unworthy of the effort.
That said, his arrest does seem to be an overreach of the church's authority.