Jonathan Chait makes a really good point:

The idea that it is important to safeguard the dignity of the presidency is one of those ideas, like “reducing deficits is always good,” that’s shared so widely within official Washington that it is considered a bland truism rather than a point of view. ... President Obama’s appearance on “Between Two Ferns” with Zach Galifianakis has prompted another round of worried harrumphing ...

Is this apparatus really too weak? Why is it the role of the press to worry that the president is coming across too much like an equal citizen and not enough like a monarch? Washington’s dignity fetish is one of those manifestations of the cult of the presidency that expresses some really weird ideas about how democracy is supposed to work.

For what it's worth, I thought the president came off unscathed. The segment was actually pretty funny, if far from Galifianakis's best.