It depends on what you call paperless. In my last office we did virtually everything by e-mail, fax, etc. However, there are certain forms that have to be filled out manually. There are some companies that will not accept certain correspondence unless it is on paper. If your office is not using electronic payroll, there will be paper checks and paper W-2 forms.
However, the biggest issue is the mail. Despite e-mail and all other forms of electronic communications, various publications, notices, billing statements come in via regular mail. Car titles and proof of insurance have to be maintained on paper and receipts for expense accounts have to be on paper.
Then you have people, who will print a 20 page document on the copying machine, so he can lean back in his chair and read it instead of having to look at the computer screen.
Finally, please, I would appreciate it if no one told me that you should never handle a piece of paper more than twice. I use to open the bills. That was one time. I would examine the bills, that was two times. I would give them to the bookkeeper to pay, that was three times, she would cut the checks and attache the bills so I could review them, that makes four times. Some would say we should had paid all bills electronically--but everyone does not want that. With our system, each invoiced was examined three times. We had very few errors.
Finally, I remember back in the Cater years when the federal government passed the paperwork reduction act. It was about a 60 page bill. I received four copies from four different federal agencies.