If the paper doesn't have ink on it, it should be fine, however, it won't do a thing for odor. The least expensive (and, imo, the most effective) litter is wood pellets -- not shavings, but pellets. Depending on where you live, a 40lb bag of woodstove pellets cost about $4. It lasts a long time. I believe with 2 rabbits, one bag of pellets could last 2 months! Pretty cheap!
This page explains how to use them.
If bunny is eating lots of hay, then cardboard should not be a problem. Some rabbits chew the boxes (again, no colored ink) but they don't ingest the pieces. They just leave the pieces once they are torn off the box. Most of my rabbits actually ate (ingested) the cardboard - and lots of it. But because they eat plenty of hay, there were no blockages. I have heard of rabbits getting a gut blockage from cardboard, but only those rabbits that ate no hay (or very little hay). [There was a time when hay was not considered an essential part of a rabbit's diet. Cardboard, then, was rightly considered a danger because without that hay fiber, there was nothing to prevent a blockage.]