This post applies to CebPac [Cebu Pacific] flights only. Content is based on personal experience.
Yes. It is possible to check in without a printed itinerary. Actually, they can verify the booking with just a valid ID.
I haven’t experienced this personally, but a friend did in the past. She forgot to print her itinerary and we paid her travel tax using the email confirmation [PDF]. The email was sent to my account, so she used my phone to present it.
During check-in, the lady at the counter said it’s OK even if we don’t present the email. She just pulled up the record using my friend’s info, and voila! she was able to check-in hassle-free.