This is a complex subject... and it's been covered in other threads, but here's some background:
There are multiple issues which are all getting conflated into one...
Firstly, Newer Amazon Fire TV devices *fully* support native frame rate switching - but the NetFlix App on Fire TV does not. NetFlix need to add support to their App, and so far they have not done this on the Amazon platform.
There's some confusion here about 24 frames or 23.967 Hz refresh. Fractional frame rates are a consequence of some fudging that had to be done because the American NTSC broadcast (analogue) TV system was refreshed at 59.97 Hz and not exactly 60 Hz. The change to 59.97 Hz was made to facility colour broadcasts in the 1950s. For legacy and compatibility reasons 59.97 Hz is still used today on digital platforms.
The vast majority of movies are encoded at 23.967 for home distribution as this matches the 59.97 Hz rate. However, some movies are encoded at exactly 24 frames per second and blu-ray and other systems accommodate both.
Some older Onkyo AVRs (such as the TX-NR808) used to have a Faroudja DCDi processor which couldn't deal with 23.967 Hz properly - this caused a very noticeable video stutter every 41 seconds (audio wasn't affected).
Lip-sync issues occur because the video and audio streams are split, decompressed and processed separately, and then re-combined for output - but there's no mechanism to ensure the video and audio always remain exactly in sync during output. Sometimes fast forwarding or reversing will cause the problem. This issue still occurs, but 23.967 vs 24 isn't the reason for it.
If you're seeing video stutter which repeats in a completely predictable way for the duration of the movie, then this *may* be a 23.967 vs 24 Hz problem. If you 'lock' or force a device to output at exactly 24 Hz and the video is encoded at 23.967, then this will be the effect you're seeing. Most of the time it's actually very difficult to truly differentiate between the two as 24 Hz is often used generically to include both 24 Hz and 23.967. They are only 0.1 of a second different.
Another source of frustration is 50 Hz UK/European content that gets converted to 59.97/60 Hz for output via various streaming devices and apps. It looks horrible.
BBC iPlayer on Amazon Fire TV outputs correctly and uses the native rate of 25 frames per second (50 Hz) but a BBC programme (for example) streamed via NetFlix on the Fire TV converts it to 60 Hz with all the finesse of a bull in a china shop.
Regards,
James.