don't do me any favours = I don't want you to do me any favours. You don't want that person to do anything for you unless you specifically ask for his/her help.
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Means the favors are mere perception and not really favors in reality.
When the undertaker begs the godfather for justice he is aware that one day,that day may
Never come he'll be obliged to return the favor. Don't do me no favors...omerta.