Hi jbeerling,
I understand you want to turn off subtitles for your Netflix episodes. I'm including the User Guide for Apple TV (4th generation) for you below with instructions for how to customize the settings for subtitles.
http://help.apple.com/appletv/#/atvb5ca42eb9
If you prefer special accessible captions, such as subtitles for the deaf and hard of hearing (SDH), you can set Apple TV to list them instead of standard subtitles and captions. You can also customize the look of subtitles and captions.
Note: Not all videos include closed captions.
Turn on Closed Captions and SDH. In Settings , go to General > Accessibility > then turn on “Closed Captions and SDH.” Or, use the Accessibility Shortcut. See Accessibility Shortcut.
Customize your subtitles and captions. In Settings , go to General > Accessibility > Style, where you can choose an existing caption style or create a new style based on your choice of:
Netflix has some additional information relative to their content. Here is the link to their website:
https://help.netflix.com/en/node/372
Happy Holidays!