I was under the impression that IPS LCD displays were highly unsusceptible to burn-in. Am I mistaken?
Yes...and no. It's a little bit different than OLEDs, which will start to show age since each pixel provides its own light source, and the different colors will degrade at different rates over time (the blue OLEDs will start to lose their brightness quickest)...
LCDs do suffer from image retention.... the pixels can't get back to their relaxed state if they were held in the same state for too long... the effect is usually temporary but can become permanent... I've seen phones with LCD displays that had permanent retention of stuff like the status bar icons, or a demo display phone, etc. Other than that... the backlight can start to degrade a bit, but you wouldn't really notice that as much since there is a common backlight. Any display, LCD or OLED, if it's showing the same thing all the time, will start to show some stuff at some point.
I wouldn't worry about IR on the second display though.... after all, that's the ONLY purpose that it serves. You won't be watching a youtube video on it or anything of the sort.