The newspapers (printed editions) are dying-circulation drops every year.
Printed books are being replaced by E-Books.
E-Mail is doing away with letters, envelopes, copies, etc.
With this drop in demand, is it logical to assume that the paper industry will start to shrink (drastically)?
The printing paper industry has been shrinking for quite a few years already.
However, only about a third of the world’s paper consumption is printing paper, the other two large categories are packaging and hygiene. In packaging, paper and fiber products are holding their position compared to plastic, and new products are even trying to displace plastic for some applications (for an example, note that the EPS molded inner packaging for electronic thingamajigs has almost disappeared - at least here in Europe - and has been replaced by board and/or molded fiber). The other large category, hygiene, i.e. tissue and toilet rolls, doesn’t really have any non-paper competitor products. I guess it’s still gonna take a few years before we can wipe our butts with a computer or an iPad 
The biggest problem for Western paper industry today is the industrialization of developing countries who now can provide an adequate commodity product at a fraction of the personnel costs we have in the first world.
A lot of pulp/paper manufacturing long agho departed from the northeast. A lot of the forests in NH have been converted to ski areas and the locomotives used to haul logs now carry tourists.
I interviewed for a job in Massachusetts in a repurposed timber industry building
McDonalds and Burger King are still using paper wrappers and are back to paper boxes. And now the Whopper comes in both a box AND a wrapper.
Paper will never go out of business.
I work with a lot of people who have been laid off from the pulp/paper/wood industry over the last 3 years, or so.
My dad worked in the pulp/paper industry for 40 years. He semi-retired about 15 years ago when the plant he worked in for 25 years started closing up shop. It’s actually still running, but barely. The little town where we lived would fold ten minutes after that plant went out of business. He ended up working for a different plant (within the same corporate umbrella) in a different state for the nest 15 years, as a consultant, mostly … and that plant is on its last legs as well.
Yes … layoffs are de rigueur in the paper industry these days.
My big brother travels around the world helping to start new paper factories, mostly in South America, China and Russia. It’s not that there’s no need for paper any more, it’s just one more industry that can be relocated to countries with cheaper labor and cheaper raw materials nearby.
Here in Finland the industry isn’t doing that well, on the other hand.