Does a vice president retain their tie breaking vote in the senate during an impeachment trial if it is the vice president being impeached?
It depends, in part, on whether the vice president is acting president at the time of impeachment.
Though not mentioned in the Constitution, Senate impeachment trial Rule IV provides that the Chief Justice presides when the vice president, as acting president, is impeached.
When the President of the United States or the Vice President of the United States, upon whom the powers and duties of the Office of President shall have devolved, shall be impeached, the Chief Justice of the United States shall preside; ...
In that case, the Chief Justice will preside and cast tie breaking votes.
When the vice president is impeached in any other case it is unclear whether the vice president can preside at their own impeachment trial. In Can the Vice President Preside at His Own Impeachment Trial?: A Critique of Bare Textualism, Professor Joel K. Goldstein critiques the argument given by Professor Michael Stokes Paulsen.
Professor Michael Stokes Paulsen recently imagined a scenario like this, examined the text of the Constitution, and, in a clever article, suggested that the Constitution would permit the Vice President of the United States to preside over his own impeachment trial. "Now that’s stupid," concluded Professor Paulsen, in nominating this result for a coveted spot on the list of "Constitutional Stupidities." Professor Paulsen is not the first prominent scholar to express this conclusion. In 1983, Professor Stephen L. Carter concluded "that the Vice President could preside at his own impeachment trial, should he choose to do so," a conclusion he described as one of "a few glaring errors" the founders made. Seven years later, he confidently repeated his claim that "[t]he founders designed a Constitution in which the Vice President presides at his own impeachment trial" as evidence of the interpretation of the document. Michael Gerhardt, a leading authority on impeachment, endorses this conclusion, as did Richard M. Pious, a foremost expert on the Presidency.
Were these analyses applied as given, that the vice president could preside over their own impeachment trial, then it would follow that the vice president could cast tie breaking votes for all questions except for the final "shall be impeached" question.
Goldstein then argues there are different "Modes of Interpretation" that allow for a different conclusion. Later, in "A Critique of the Vice President-Presides Thesis", he says,
Although the text does not say the Vice President cannot preside, it does not say that he can preside at his own trial either. On the contrary, the text does not speak specifically to the subject of who presides when the vice president is tried. It is, in essence, silent. Accordingly, we cannot determine whether [the vice president] can preside over his own trial simply by reading the text; we must invoke other tools of constitutional analysis.
After analysis of the framers' actions during the writing of the Constitution, Goldstein says,
To the extent Professor Paulsen believes that the “omission of any such exception [to the Vice President presiding over his impeachment] can scarcely have been accidental,” he is, I believe, mistaken. It is inconceivable to imagine James Madison and the other founders consciously considering with equanimity the prospect of an impeached Vice President presiding over his own trial and concluding that it was good, especially given Madison’s opposition to self-judging. Its omission does not reflect a conscious intent to distinguish between the situation of the President and the Vice President. Had they thought of it, the framers surely would have provided that either the president pro tempore or the Chief Justice would preside over the Vice President’s impeachment trial.
Or perhaps they thought they had, in fact, specifically so provided, at least for most occasions. The framers conceived the Vice President as a legislative officer. As such, it was unlikely that occasion would arise to remove him by impeachment and conviction. The time when the Vice President would be at risk would be on those occasions when he exercised the office of the President. The weight of the evidence suggests that the framers did not intend the Vice President become President upon the death, resignation, removal or inability of his predecessor; they intended him only to exercise the powers and duties of the office. Thus, the framers conceived that when the Vice President acted as President, and accordingly would be most vulnerable, he could not preside over the Senate! It would be odd to conclude that the framers intended the Chief Justice to preside when the President was tried for impeachment but not if the Vice President acting as president was. More likely, the framers’ intent was that the President pro tempore preside whenever the Vice President acted as President but that the Chief Justice would preside over the impeachment trial of the President or Vice President acting as President. The Vice President has evolved into an officer of the Executive Branch. He rarely presides over the Senate, except on ceremonial occasions or to break ties in favor of the Administration. In view of this ongoing history, it would be anomalous if he sought to preside over a substantive event as momentous as his own trial!
Given this analysis of "intent", the vice president would not preside, leaving the president pro tempore of the Senate as the presiding officer. As a senator, the president pro tempore could cast their own vote to break (or make) a tie.
In practice, the Senate resolution authorizing the impeachment trial would contain a rule regarding participation of the vice president (as presiding officer) during their impeachment trial.
Note that, by precedent,
The Vice President has no authority to decide constitutional questions. He has no rulemaking power over the Senate. The Vice President must apply the rules of the Senate as they are. [Riddick's Senate Procedure]
Thus, the vice president cannot question, on constitutional grounds, a rule adopted by the Senate prohibiting the vice president from presiding over their own impeachment trial. It follows that, if they cannot preside over their own impeachment trial, they cannot cast any tie-breaking votes.