Yes, I print everything and keep them in the order I'm going to need them. It's easier to pull the next piece of paper than to fumble for a booking e-mail from 3 months ago or remember whether this hotel was on booking.com or hotels.com or the hotel site. I want to have it ready when I approach the counter in case I need it, and if the receptionist needs to see a confirmation, I want to hand over a piece of paper instead of my phone.
Seth