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.]
Generally, newspapers are printed with a vegetable based ink and are totally safe--been using shredded paper in boxes for 15 years + with no problems and they love playing in it too. Same is true with cardboard boxes--the ones you want to avoid are those that have photo quality pictures as they are a plastic applique and definitely not good but ones with inked surfaces from inside the U S are also done with vegetable ink, probably due to our "sue" over anything society and EPA mandates. There was a whole bunch of box uses under "Cardboard Castles" here once upon a time. I used to have a mini Rex that would get into a big pan I had filled with hand shredded newspaper and she would "submarine" under the paper and move all around and then "prairie dog" and pop up through the paper and look around--funny to watch and she'd entertain herself for hours.
Call and ask or ask your Vet-but since a lot of us wet a finger to get a better grip to turn to the next page, if it wasn't safe my widow would own several news companies and we've used it with our bunnies for almost 2 decades now.