Grammatically speaking, in standard/formal English a negative negates something. Two negatives together form something called a double negative, where the second negative reverses the first one. This is not the case in all languages, but it is in formal/standard English.
The sentence you propose means that you are [no longer] [not hungry], which means that you were "not hungry", but this is no longer the case (so you are hungry now). Also, "following the rules", the example provided by @SovereignSun in comments "We don't need no education" actually means "We [do not need] [a lack of education]", i.e. "we are open to getting some education". Obviously, the intended meaning is "We don't need any education". I suspect this may be part of the protest message of the song - using "ungrammatical English" to state a dislike for rigid school rules.
Colloquially, however, double negatives are often taken to mean, or even to emphasise, the negative, especially in certain dialects. E.g. "I don't know nothing" would be taken to mean "I don't know anything", not "I know something" (which is what it means in standard/formal English). Most double negatives that you will hear employ this use of negative concord to mean or emphasise a negative meaning. However, as this is a colloquial meaning, it should not be used in formal English, and can make you sound "uneducated" to some ears.
Just to add extra confusion, there are times where double negatives are used grammatically to good effect, where everyone would understand you (especially spoken with appropriate emphasis). E.g. "I couldn't not help him", which means I had to help him.
As is the case with many things, context is required to understand the meaning of a double negative. A good example of this would be the phrase "can't do nothing", which would rely entirely on context for you to understand the intended meaning. It could be used either as a positive:
We can't do nothing! We've got to do something to help!
Or (colloquially) as a negative:
It's broken, I can't do nothing to fix it.
Because of this potential for confusion (especially out of context), and because negative concord is used colloquially, but not formally, I would in general not recommend using double negatives.
So therefore, you can use "I am not hungry no more" and people will understand what you mean, but saying "I am not hungry any more" is more grammatically correct, and is the form you should use in formal prose.