I print my boarding passes at home (or, more likely, my office) in case my phone is dead, some scanner in the airport is on the fritz, the TSA (or the foreign equivalent) decides it has made up new rules, I'm traveling somewhere I haven't before and don't know for sure if my phone can find the network there, etc. I can't count how many times someone's phone has caused problems for someone in front of me in some line.

Travel as you wish, but I always have full paper documentation of my travel plans with me, including boarding passes. I use my printed notes for a given reservation (flights, hotel, car, etc) to take notes on conversations that I have with CS for those reservations. Having that information has been valuable a multitude of times --- one's assertions are given more weight by being able to say, "when I spoke with Daniella on January 12th, I was told ..."

Again, travel as you wish, but I do not see a future where I don't have printed documents with me on a trip.