- Not any movies won't be playing on Christmas day.
You're using a double negation here, so you're basically saying that there is not a single movie that will not be shown on Christmas day, which is the same as saying that all movies will be playing on Christmas day. Rephrase your sentence like so:
- Not any movies will be playing on Christmas day.
This sentence would probably be more idiomatic if you changed not any to no:
- No movies will be playing on Christmas day.
In the second sentence, you also have a double negation:
- Not any movies won't be being shown on Christmas day.
...won't be being shown is syntactically correct but sounds off and, to be honest, I have never heard it being used by anyone. You can leave out one auxiliary and retain the meaning. The being is obsolete here, because it focuses on duration. However, since no movie will be shown at all, there is no need to focus on the duration of the showing:
- No movies will be shown on Christmas day.
- There are no movies playing on Christmas day.
- There will be no movies playing on Christmas day.
These sentences are both correct. Sentence 3 uses a simple present (are) to express a future, which is perfectly acceptable when talking about things that are fixed, such as time-tables, schedules or in this case the programmation.
Sentence 4 uses a simple future and is also a correct sentence.