As tharne mentioned in another comment, this isn't really true. Harvard (like all private universities) gets tons of government funding: directly via grants and indirectly via the federal loans that a lot of its students get. Society therefore arguably gets to have some say in how Harvard does business; if Harvard doesn't like that, it can stop taking federal grant money.
Add on to that the massive tax break they get on they endowment's huge investment earnings.
Harvard is a hedge fund with a school attached. They should be taxed like one.
Private institutions are bound by laws that we (society) create.