The issue here has nothing to do with Christianity. If you text your girlfriend and tell her you are going to burn her, that is a very specific threat from one individual to another. You may get a visit from the cops but I don't believe they can arrest you for that. It's probably good grounds for a restraining order though. Holding a sign saying that 'gays will burn in hell' for example is a very general threat which is not against any particular individual and it doesn't state the sign holder will do the burning. It is clearly different than 'I will burn Ellen DeGeneres".

Now if particular speech can be shown to incite violence then there may be a criminal action there, but as shown by many white supremacists, that is pretty hard to prove.