Why do some people believe that life is a zero-sum game and that if you harm someone, you will benefit? I have a strong feeling that many people consider life as a zero-sum game, i.e. they always assume that you'll benefit, if you harm/disadvantage someone. Any research on this?
There is definitely more than one aspect to this question. One of them is if there is a certain type of person that views life as a zero-sum game. What I looked into is if there is a certain type of person that actually gains something from hurting others. 
There seems to be: The everyday sadist. 
Buckels, Jones and Paulhus (2013) mean that there is a sadist personality type that is expressed in everyday life. (See: http://pss.sagepub.com/content/24/11/2201 and http://www.psychologicalscience.org/index.php/news/releases/everyday-sadists-take-pleasure-in-others-pain.html for an article summing up the scientific article).
These researchers have been looking into traits that they call the "Dark Triad". "When aggression was easy, sadism and Dark Triad measures predicted unprovoked aggression. However, only sadists were willing to work for the opportunity to hurt an innocent person." 
People strong in this trait would probably feel affirmed by a world view that enables them to fund pleasure with a rationale behind it.
This answer is inadequate as a complete response to your question. 
Something I'd like to look into further is the Just World hypothesis, where the belief that the world should mete our justice leads us to act in a certain way.