someone who always uses an excessive amount of paper. If they're writing an essay in pen and make a mistake, they ball up the paper and take out a new sheet instead of just crossing it out. If they're drawing something and they mess up, they take out a new sheet instead of just using the eraser. You get the idea. The opposite of a tree killer would be a tree hugger, one who is very frugal in their paper usage.
Just imagine this: if your teacher gives 10 papers a week to each student and there's 20 students in your class, that's 10 papers x 20 students x 7 class periods x 36 weeks of school. That's 50,400 papers used by that 1 teacher every school year. What a tree killer!