There are a variety of estimates of varying validity for total world paper usage. I'll stick with US paper usage, which is 69 million tons per year, according to the EPA (Note, I have no idea if the linked site is no longer maintained due to old data or due to change in political climate. I will just have to assume it is correct). From here, the most reasonable world estimate I found is 336 million tons, which seems about right.
According to a 2007 FAO report (page 7) paper and pulpwood consumes about 550 million m3 of wood per year. Somewhat surprisingly, this matches the world estimate above, since 336 million tons per 550 million m3 works out to 610 kg per m3, which is right between average densities for hardwoods and softwoods.
So now that we are more confident on how much wood is being used, we have to see how many trees that is. Timber yields per hectare vary wildly, as do yields per tree. I can't put together any reasonable estimates there. However, we can see from the FAO report that the 550 million m3 accounts for about 35% of the total 1550 million m3 of industrial wood production, which (from a different FAO powerpoint, page 9) itself only accounts for half of total timber clearings (the other half being firewood and agricultural clearance such as slash and burn).
However, the final thing to note is from the first FAO report. There it is shown (page 6) that for pulpwood for paper specifically, around 95% of demand is met from plantations.
Now, let's take a look at deforestation. Deforestation (page 2 of second FAO link) is fastest in parts of the world where more of the wood is being used for firewood and land clearance (Africa, South America, Central America). Wood pulp itself is mostly made in North America and Europe (FAO forestry products Yearbook 2014, page 224, I can't figure out what units those are either). NA and Europe combine for 66% of world production. Looking at our other sources, NA and Europe are seeing net forest gains, and so the pulpwood creation there is probably as sustainable as we could ask for.
World pulpwood for paper production appears to be sustainable. The evidence is that the majority of pulpwood production comes from areas of the world with increasing forest areas, and the vast majority of pulpwood production comes from plantations.
Furthermore, the areas of the world with higher deforestation are associated with high firewood consumption and land clearance for agriculture. Thus, shifting pulpwood plantations to other wood uses, like sawn roundwood (for construction), is not likely to help ameliorate deforestation.
All in all, the best thing for decreasing deforestation would be to shift the energy usage in poorer countries from firewood to electricity and cooking gas, and increase mechanical agriculture to increase crop yields. Of course, both of these things will inevitably increase fossil fuel consumption instead. Catch 22.