How many Senators must vote to override specific judgments of the Chief Justice in an impeachment trial of the President?
A simple majority.
During the impeachment trial of Andrew Johnson, there were several examples of the Senate voting on questions of relevancy, materiality, and redundancy of evidence and incidental questions, which were all decided by a majority. It should be noted that much of the current rule 7 was developed during the trial and at least one issue included a vote by the Chief Justice to break a tie.
Had you on the evening before seen General Thomas? * * * Had you a communication with him?
Answer. Yes sir.
Mr. Stanbery objected, and the Chief Justice ruled that the testimony was competent and would be heard "unless the Senate think otherwise."
To this ruling Mr. Drake objected and appealed from the decision of the Chair to the Senate. It appeared to be not to the ruling per se, that Mr. Drake objected, but to the right of the Chair to rule at all upon the admissibility of testimony. Mr. Drake representing the extremists of the dominant side of the Chamber. There seemed to be apprehension of the effect upon the Senate of the absolute judicial fairness of the rulings of the Chief Justice, and the great weight they would naturally have, coming from so just and eminent a jurist. After discussion, Mr. Wilson moved that the Senate retire for consultation.
The vote on this motion was a tie, being twenty-five for and twenty-five against retiring, whereupon the Chief Justice announced the fact of a tie and voted "yea;" and the Senate retired to its consultation room, where, after discussion and repeated suggestions of amendment to the rules, the following resolution was offered by Mr. Henderson:
Resolved, That rule 7 be amended by substituting therefor the following:
The presiding officer of the Senate shall direct all necessary preparations in the Senate Chamber, and the presiding officer in the trial shall direct all the forms of proceeding while the Senate are sitting for the purpose of trying an impeachment, and all forms during the trial not otherwise provided for. And the presiding officer on the trial may rule all questions of of evidence and incidental questions, which ruling shall stand as the judgment of the Senate, unless some member of the Senate shall ask that a formal vote be taken thereon, in which case it shall be submitted to the Senate for decision; or he may, at his option, in the first instance, submit any such question to a vote of the members of the Senate.
The amendment by Mr. Henderson was adopted.
Amendments attempting to prevent the Chief Justice from ruling on questions of law and from voting on ties were rejected.
During the impeachment trial of William Jefferson Clinton there were a few motions to suspend the rules. A Motion to Suspend the Rules requires a two-thirds vote. Such motions are intended to alter the procedures used by the Senate. See Senate Rules XXII 2 "... except on a measure or motion to amend the Senate rules, in which case the necessary affirmative vote shall be two-thirds of the Senators present and voting ...".
Article I, Section 3
The Senate shall have the sole Power to try all Impeachments. When sitting for that Purpose, they shall be on Oath or Affirmation. When the President of the United States is tried, the Chief Justice shall preside: And no Person shall be convicted without the Concurrence of two thirds of the Members present.
In summary, for impeachment trials of the president, it requires two-thirds of the Senators present to vote for conviction and for any procedural changes that relate to rules; otherwise, a simple majority, which may include a vote by the Chief Justice, in the case of a tie.