I have various experiences with cardboard. We use it in the house for lighting fires in a conventional stove, but I also use it regularly in my smoker for my
bees as fuel. After a session the cardboard leaves a substantial amount of ash, which makes it good for the smoker (the fuel burns slowly) but less good for a situation where you want full and fast combustion. For comparison, when I use half and half cardboard with compressed woodpellets I get very little ash from the pellets and lots from cardboard.
I suspect that some of the additives in the cardboard include clay compounds - certainly the case with most papers - which explains where the ash comes from.