Why does not aqua regia burn filter paper?
For "burning" of paper by sulphuric acid, the better term would be "aggressive dehydration".

Note that aqua regia is rather a diluted acidic mixture without significant dehydrating or oxygen based oxidation behaviour, but manifests aggressive chloration behaviour, that dissolves precise metals.

Cellulose has quite good endurance against chlorine, which is used in various solutions for cellulose whitening.