Apparently there is a movement underway in New Jersey to recall Senator Robert Menendez. Eugene Volokh analyzes the case on the grounds that the Constitution does not allow for a state to recall a sitting Senator.
New Jersey law authorizes such recalls, but it seems likely that the U.S. Constitution does not allow a state to force the removal of a senator based on a recall vote. The Seventeenth Amendment provides that the Senators shall be “elected by the people thereof, for six years,” and the Constitution provides for no means by which this term can be shortened by the voters of a state. A state can no more force the removal of a Senator by recall than it can force its Senators to leave office after four years instead of six. (For more, see this Congressional Research Service report on the subject.)
The argument would be this: There should be nothing unconstitutional about the voters’ expressing their views that someone should resign, even if the expression is done through the formal process of a state-organized vote. The only unconstitutional thing would be to try to force the Senator to give up his office. So the signature gathering and election (if enough signatures are gathered) should be constitutionally permissible, and should have the maximal constitutionally permissible effect — which could indeed be a significant political effect, even if not a morally binding one.
As always, an interesting question and an interesting read, at least for those more legally inclined.