In your example, you can say either "I don't have a place for you" or "I have no place for you". These both say basically the same thing just with a different structure. You use "don't" only when negating something, in this case the verb "to have". When you say "I have no place for you" you are not negating the verb to have. Rather, you are saying that you have 0 of something. One other example could be the difference between "I didn't buy anything" and "I bought nothing". In the first case, you're negating the verb. In the second, you are saying what you did (or in this case did not) buy.

Use "don't" when negating verbs (although there are some exceptions with a few verbs such as "be" and modal verbs), "not" when negating other parts of speech, and "no" to indicate lack of something ("no vacancy") or prohibition to do something ("no littering"). That's just a rough guide.