Barney's jaw is operated by the performer in the suit. It is not radio controlled. Believe it or not, he uses his jaw to push down on a control bar that moves the mouth. It is an incredibly unsophisticated method that requires enormous neck strength of the suit performer. The production tried on several occasions to change how it worked, but the suit performer got used to it and preferred it. And Barney does not have radio controlled eyes - at first, they were mounted on swivels so they could swing on their own from side to side as the performer moved his head. Later, they were changed to a fixed position because they just jittered around too much.Barney and Welcome to Pooh's Corner, were both mechanical puppetry. Off stage puppeteers controlled the blinking eyes and mouth for the characters
That's classic peak popularity late 90's Barney, anyway. Perhaps there is more recent stuff that uses more sophisticated technology that I don't know about.