There are several well-known games in which the pareto optima and Nash equilibria are disjoint sets.
The most famous is probably the prisoner's dilemma.  Races to the bottom or tragedies of the commons typically have this feature as well.

I proposed calling these inefficient games.  More generally, games where the sets of pareto optima and Nash equilibria are distinct (but not disjoint), such as a stag hunt could be called potentially inefficient games.

It seems worthwhile to study (potentially) inefficient games as a class and see what can be discovered about them, but I don't know of any such work (pointers welcome!)

We know that Tit-for-Tat and variants do very well in iterated-Prisoner's-Dilemma tournaments. However, such tournaments are a bit unrealistic in that they give the agents instant and complete information about each other's actions. What if this signal is obscured? Suppose, for example, that if I press "Cooperate", there is a small chance that my action is reported to you as "Defect", presumably causing you to retaliate; and conversely, if I press "Defect" there is a chance that you see "Cooperate", thus letting me get away with cheating. Does this affect the optimal strategy? Does the probability of getting wrong information matter? What if it is asymmetric, ie P(observe C | actual D) != P(Observe D | actual C)?