There was a little ambiguity in the first question. "Anymore" can't be used together with "never" in the future. "Anymore" can also be written as two separate words: "any more".

You can say:

I won't do it anymore

To refer to an action that has likely been repeated. The best thing to think about is "Was this perhaps something the person did regularly, and they have now realised they were doing it wrong?"

"You've got to stop shouting and raging every time your team loses!"

"OK! I won't do it any more."

This means, you won't do it again. So, you can also say:

"OK! I'll never do it again."

But you can't use "never" and "any more" together using "will". Also consider that the future "will" or "will not" are almost certainly used in the contracted form when speaking in this type of sentence (I'll/I won't).