Over at Mental Floss, an article describes the research done by a couple of students at the University of Leicester about how much paper it would take to print the entire internet.
Their answer, based on the number of pages indexed by search engines, is somewhere between 68 billion and 136 billion pages, which would require about 44 square miles of Amazonian rain forest to make. If, that is, paper was made from Amazonian rain forest trees instead of the managed timberland which is actually used.
Now, if we're just talking about printing the useful parts of the internet, I suppose we could get away with taking one or two limbs off of a tree. Which would probably heal itself pretty quickly if we used the rest of the internet for its most suitable function of fertilizing said tree. I leave the decision of which group would contain this here blog totally up to you, O Perceptive Reader.