Update: as of 2023, RyanAir no longer issues boarding passes to non-EU/EEA passport holders on flights which cross Schengen borders at all, so you do need to go to the check-in desk to get your boarding pass. Thus the question no longer applies.
Below remains just for reference:
For the last several years I flew RyanAir, I have never had my boarding pass stamped (I have non-EU/EEA passport). I have never been refused boarding, and never seen anyone being refused boarding.
The typical RyanAir procedure depends on airport, and is one of the following:
They scan everyone's non-EU passport regardless of whether one did check or not. This is typical at BER airport.
They call non-EU passport holders who didn't get visa check to the gate agent before boarding, and scan your passport there.
They walk along the line ready for boarding, and ask who has non-EU passports and didn't do check, sending us to the gate agent for the check.
Note that this check takes some time (10-30 seconds), and they do close the gate 30 minutes before departure; if you do so, try to arrive earlier than 30 minutes before the gate closure time.