If the House votes to impeach President Trump on grounds not authorized by the Constitution, its action, in the words of Hamilton, is void. As he put it in the Federalist Papers, “no legislative act, therefore, contrary to the Constitution, can be valid.” If this is indeed the case, then the Senate will be confronted with a constitutional dilemma, if and when it will receive a void and invalid impeachment. It will have to decide whether to proceed with a trial of charges that are unconstitutional and therefore are void.
An analogy to consider from ordinary criminal cases may be imperfect but informative. If a grand jury were to indict a citizen on an unconstitutional “crime,” like marrying a person of a different race, the trial judge would immediately dismiss the indictment and refuse to subject the defendant to a trial. Indeed, the House plays a role similar to that of a grand jury in the impeachment context, and the Senate plays a role similar to the trial court. In the presidential impeachment context, the chief justice of the Supreme Court presides and rules on the legal and evidentiary issues.