Anymore is one word, while any more is a phrase—and they have different meanings.
The phrase any more is simply a combination of the words any and more, in which each word has its independent meaning, and is used to modify nouns, where it means the same thing as “more,” though the word any is used mainly in negative contexts. Some examples will clarify the point:
Correct: I don’t want to eat any more meat.
Correct: I do want to eat more meat.
Incorrect: *I do want to eat any more meat.
However, any and any more can be used in some contexts without negation, such as at the beginning of sentences, in subjunctive contexts, and in questions:
Correct: Any more people, and the room will be too crowded.
Correct: I would welcome any more suggestions.
Correct: Do you want any more (food) to eat?
Incorrect: *Yes, I want any more (food) to eat.
Anymore, on the other hand, is an adverb which means (not) “any longer” or (not) “for more time.” Like any, anymore is used mainly within the scope of negation and questions:
Correct: I cannot sing anymore; my voice is tired.
Incorrect: *I can sing anymore; my voice feels great.
Correct: Do you go running anymore, or have you stopped?
Incorrect: *I do go running anymore.
Notice that any more and anymore can both be used after verbs, but with slightly different meanings:
I will not sing anymore. = I will never sing again.
versus...
Can you sing another folk song? No, I will not sing any more (folk songs).
So be careful to think about which idea you are trying to convey:
anymore = more of the action / any more = more of some noun.
"Not Anymore" Meaning
And finally, English learners often ask about the phrase not anymore, as it is a common and useful English phrase. The meaning is apparent in the examples above, but just to reiterate:
The phrase not anymore means that an action or thing no longer continues or no longer exists. For example, if you say "I'm not a smoker anymore," you mean that you used to smoke in the past, but now you don't. The phrase is often used to mark a transition, such as a change in status:
I am not a child anymore
I was the number one salesperson in the region, but not anymore.
And changes in thought or feeling:
I used to think Alice was the best actress in Hollywood, but not anymore.
I do not like drinking coffee anymore because it makes me feel jittery.