It happens even in very serious criminal trials, where the defense argument is basically: "The prosecution clearly hasn't met its burden of proof." And they make a closing argument without ever presenting a case.

IIRC, the defense for Michael Lee King in the murder of Denise Amber Lee (a death penalty case, no less!) didn't present a defense, but during the penalty phase they did call character witnesses in mitigation. Of course it is their right not to mount a defense, but it always blows my mind when it happens, especially in a capital case. And the jury is not allowed to draw any negative inference from it.

Of course this is different because it's an impeachment trial in front of the senate and the legal parameters are different.