I heard that the screen protectors might peel off the anti-reflective coating from the screen. Is this true?
Depends on what was used for the screen protector to adhere to the screen.
I have used screen protectors on all my iPads (and most of my iPhones up to the 5-series) and never encountered any issue. Having said that, once a screen protector is on, I have never removed it unless to put another one straight back on.
Also, I use the plastic-type screen protectors which use some kind of electrostatic or some other magic airy-fairy stuff to stick to the screen and the screen protectors do come off pretty easily without any residue, thus leaving me to believe that the coating (oleophobic or anti-reflective) has *not* come off the screen.
Have not used glass or other hard screen protectors which might need stronger "glues" thus I cannot comment on those.
TL;DR - used plastic screen protector, no issue.