Question via Instapundit, via Digg.
It would seem the Constitution doesn’t provide another option, unless “being impeached” counts as “exercising the office of President of the United States,” which I can’t imagine it does.
He most certainly could. But what purpose would it serve? The presiding officer at an impeachment hearing lacks the power of a true trial judge, because any decision they make can be overruled by a majority vote. So if the Senate was inclined to convict Cheney in the first place, his being the presiding officer would not realistically be able to help him out (since everything he did could be overruled easily) and in fact it would probably hurt his situation by making him look bad.
How could you tell?
We have raised, discussed and dismissed the possibility numerous times. Search is your friend. 
Query: V-P Cheney Impeached; He gets to preside at his own Senate trial?
So suppose Dick Cheney’s impeached - does he preside over his own Senate trial?
To name just two.
Neither the Standing Rules of the Senate nor the Senate Rules on Impeachment Trials provide for the situation, except for cases when the “Vice President of the United States, upon whom the powers and duties of the Office of President shall have devolved, shall be impeached, [in which case] the Chief Justice of the United States shall preside.” (Impeachment Rule IV)
I belive that under the Constitution the Vice President cannot be prevented from presiding over the Senate whenever he pleases. Because the Constitution specifically designates the President as the only officer over whose impeachment trial the Chief Justice presides (and the Senate Impeachment Rules expressly designates the Chief Justice as presiding officer over Vice Presidential trials only when the VP is acting as President), it would be difficult to argue that the VP cannot preside over his own impeachment trial, should he be crazy enough to do so.
However, it would most likely be an idiotic thing to do because the trial of an impeachment by the Senate is, at its core, a political trial, with Senators votes based to some greater or lesser degree on the likely political response of their constitutients. Virtually anything a Vice President would do as chair would make him look arrogant, foolish or both to Senators and the public at large. Not to say that there wouldn’t be a Vice President who might have the arrogance to do it, but it would probably would not be an advantegous strategy.