I've only encountered a few places that demanded actual paper. If I have the email confirmation, it will suffice most of the time. Any place that requires paper - usually a voucher - has made it very clear in the email.
I print paper copies for two reasons. One I have given before, and I can only say until it happens to you, you'll probably think I'm obsessing. But a bricked phone can really ruin your day.
The secondary reason I print is the same as why I keep the electronic confirmations. Not to prove anything, but as a reminder to ME of what exactly has been booked.