tl;dr Don't stick your phone in your face.
It depends on flicker and frequency...
Assessment of the effect on the human body of the flicker of OLED displays of smartphones addresses this, relying on LED Lighting Flicker and Potential Health Concerns: IEEE Standard PAR1789 Update.
The criterion recommends that the percent flicker be 5% or below at 90 Hz or less, because this is the luminance range at which flicker can be detected, which can cause serious symptoms such as seizures. The 5% criterion alone does not cause intense symptoms and may still cause discomfort. The standards for the flicker frequency, fflicker, have been established for invisible flicker at frequencies higher than a visible flicker.
In [IEEE Standards PAR1789], percent flicker (1) has a maximum level of 100%, and the frequency that satisfies the low-risk level is 1,250 Hz or higher. Although IEEE Standards PAR1789s recommended level is over 1,250 Hz, some experiment results have revealed that in special cases, flicker can be recognized at about 3 kHz.
"Flicker" is the difference in luminescence between the max and min states. 100% is when the LED turns off. 50% is when it dims by 50%. And so on.
If your PWM flicker is 100% it needs to be at least 1250 Hz to be "low risk" and 3000 Hz to have no effect.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/EDsFNQ9Z.jpg)
...but diminishes with distance.
However the same paper found that the visibility of flicker diminished with distance and with higher brightness. The graphs below are tests with an OLED smart phone using PWM dimming at 240 Hz.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/GEm6etQE.jpg)
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/bfFAkkUr.jpg)
Here the voltage received by the photodetector is a stand in for flicker perceived by the human eye. In both cases it was imperceptible at about 5 cm.
Maybe they need reading glasses.
The effect is exacerbated by putting your phone close to your face. I am very nearsighted, with an astigmatism and damage to my optic nerves resulting in blind spots. When my eyes get tired they have trouble focusing, so I lower my glasses and hold my phone about a hand-span away from my face to read small text.
In What is PWM display flicker, how to deal with it the author has a similar story.
In my case, one eye had astigmatism, and my optometrist was able to give me a new reading glasses prescription to correct for that astigmatism. Many people who are sensitive to flickering light and flickering smartphone displays also have astigmatism, and correcting for this problem could very well solve your sensitivity issue.
In my case, I'm able to use flickering phones so long as I wear my glasses, but even this is within a limit. I don't dare use any of these flickering phones below 50% brightness or in a dark room, and I generally cannot use them when I'm not wearing my glasses.