Netflix has started blocking rooted devices from installing their app. Currently, this only affects devices that use the Google Play Store. This means that, for the time being, Amazon devices are not affected, so rooted Fire TVs and Fire TV Sticks still have no problem installing and using Netflix.
A couple of days ago, Netflix on the Google Play Store started using Google’s Widevine DRM technology. This means that “devices that are not Google-certified or have been altered” are no longer permitted to install the Netflix app through the Google Play Store. Rooting a device or unlocking its bootloader qualifies as “altering” the device and makes it incompatible with Widevine.
Amazon’s devices, including the Fire TV and Fire TV Stick, do not use Google’s Widevine DRM technology, so the Netflix app in the Amazon appstore does not require Widevine. This is why this change does not affect rooted Fire TV devices.
It does not seem as though Netflix’s app actually checks for Widevine certification because devices unable to install Netflix from the Google Play Store, due to this change, are still able to sideload the Netflix app and use it just fine.
The new adoption of Widevine for Android devices using the Google PLay Store could mean that Netflix is going to start cracking down on rooted devices, in a similar manner to how they began cracking down on VPN use not too long ago. For the moment, everything is fine for rooted Fire TV devices, but that may change in the near future.