I've chucked paper and cardboard into biochar burns. Sometimes you can still distinguish it after the burn, because that char ends up in thin flakes. There is really nothing bad that I can think of that would be left in that kind of char. Even if you used glossy colored paper, which has a high clay content. In fact, a little clay in the biochar might be an added benefit if you were adding it to a very sandy soil.
Edited to add: What makes good biochar burn starter material are cabbage boxes. The paraffin on that cardboard board makes it light up like fatwood. So the next time you are
scrounging supermarket throwaways, cut those into pieces and use them.