Can I recycle pizza boxes?
No, because of the grease.
Greasy or food-crusted cardboard (and paper) are not recyclable because the oil impedes the paper recycling process.
To recycle paper products, manufacturers use a pulping process which mashes the paper back into cellulose fibers using water. When oil enters the stream the quality of the new paper is diminished, sometimes to the point of being worthless.
So pizza boxes are trash?
Not necessarily. Often the tops of pizza boxes remain grease free thanks to those little plastic triangles that prop them up. If you've got a clean pizza box top, tear it off and recycle it.
And the greasy parts can be composted. Tear cardboard into small pieces to speed up decomposition.
(Those little plastic triangles are not recyclable, btw, because they're just too small.)