I’m not sure what is meant by “local channels.” Most cities have local affiliates with ABC, NBC, and CBS. These affiliates carry the network shows plus local programming, including local news. There is no way that cable companies would drop those major network affiliates. The only other local station is the public access channel, which - where I live - is mostly a bunch of goofy leftists like Democracy Now! and a few conspiracy-mongers. As for televangelists, there are two major national religious networks - Daystar and TBN. I don’t see any reason why cable networks would drop them.
To correct myself, we do have a couple of other local channels - the University of Washington channel, the local government channel, and maybe one or two others.
I could be wrong, but I think by locak channels, they mean local affiliates. Those local affiliates as well as any other local TV station are what are required by law today.
If it's not required by law and the compensation set by law. Then cable channels will negotatiate and it is very possible if they don't get the deal they want, they would drop the local station.