To turn on/off closed captions for antenna channels on the Fire TV:

From the Home Screen, go to: Settings > Accessibility > Closed Caption > Closed Caption

And set it to on/off depending on what you are trying to do.

Additionally, while watching a show, you can turn the captions on/off right from the show. 

To turn on/off captions while watching a show: Press Options (three horizontal lines) > Captions and Audio > Subtitles and Captions

Change to Off/cc1 depending if you want captions off or on.