Though forests used to be cut down for paper. That's no longer the case because it's not profitable anymore. Because of lack of demand.
But while the parent poster is right that virtually all paper today comes from sustainably farmed wood. That's still an ecological problem. Because sustainably farmed wood is not native to our eco system. Brazilian flora might take centuries to grow, so farmers would chop down local trees to plant European pine trees instead. Their propaganda, which the parent poster bought into, is that if you chop a tree to plant another, then no harm is done. But that's not how eco systems work. Local fauna cannot live in foreign pine trees, just as they wouldn't live in a concrete city. So many local species are now endangered or extinct because of paper farms.
I live on an area where a lot of paper farming used to happen. I had an American friend come here once. He told me the forest around the highway we were on surprised him. Because it looked nothing like the movies he seen, instead it looked just like Europe, with regularly spaced tall pine trees. He asked "is this where the monkeys are?", looking closely to the pines hoping to see one. Unfortunately, I told him, during my whole life here I have never seen a monkey in the wild. Even though this same place used to be the home of some really interesting species, like the golden lion monkey. But we could go to the zoo to see one if he wanted to.
Source of this is my experience as a local who used to be an active Greenpeace volunteer, back in the day.
For the record, I didn't suggest that replacing old growth forest with fast-growing trees foreign to the ecosystems is neutral or sustainable.
Most paper you can get is made from tree that are grown specifically to be made into paper (http://en.wikipedia.org/wiki/Tree_farm) - those are managed and replanted just like any other kind of farm. They are also brilliant CO2 sinks (as long as the paper isn't burned). Look for any of the organisations mentioned in the WP article -- ATFS, FSC or SFI -- next time you're in an office supply store in the US, similar schemes for other parts of the world. You'll be hard pressed to find paper that isn't certified as being sustainably farmed under one of these schemes.
I can't speak for Russia, but the disappearing old growth rain forests in among others Brazil isn't for paper production, it's getting cleared for farming or harvested for the exoticness of the wood (for furniture etc) - not for paper production.
To be fair, I was only considering the fiber-end of the equation. Processing fiber (from wood or recycled paper) is energy, and (in particular) water intensive.