During Shacharit (on Shoshana Purim) the Rabbi at the shul where I daven began to say למנצח, and then I publicly corrected him. Should I ask for מחילה from him or is there no need?
I understand your question that the Rabbi was davening as the chazzan and he made a mistake and you corrected it. The Rabbi surely doesn’t have any ill feeling towards you, because you did what you were supposed to do. In fact you did him a favor and corrected him so everyone will now do what is right. Besides everyone knows that human make errors.
However it would in any case be a nice gesture to apologize to him, but say it in passing without making it into an issue, because you don’t want to make an issue over something that we might be embarrassed about. If you think that he is the type of person that mentioning it will embarrass him, then don’t say anything.