An OLED monitor has been on my wishlist since forever, but I'm not about to drop $800–$1,000 on one anytime soon. Fortunately, I got to experience a fantastic OLED display that I recently received for testing. I spent a few days on the display, playing games and working a bit, just to see what having an OLED monitor felt like.
What I also did was ignore the regular pixel refresh prompts on the screen until I couldn't skip them anymore. The monitor decided to spring the pixel refresh cycle on me when I was enjoying a particularly engaging segment of my Deus Ex: Human Revolution rerun. If you just bought an OLED monitor or are planning to get one soon, don't skip the regular OLED maintenance when your monitor asks you to do it.
I skipped the pixel refresh cycle one too many times
Turns out I didn't have unlimited lives
Since I was only testing this particular OLED monitor for a few days, I didn't really care about skipping a few pixel refresh cycles. It's not as if I was running it 24/7 for a week — I would have spent a total of under 20 hours on it. I was most interested in comparing the pristine OLED image quality against that of my 5-year-old IPS monitor. And ignoring the monitor's repeated prompts to run the pixel refresh cycle for a few days wouldn't have "damaged" the display in any way.
However, the one downside to saying "maybe later" three or four times in a row is that your OLED monitor will force you to run the cycle at the most inopportune time. In my case, it was when I was running from Barrett in Deus Ex: Human Revolution, desperately low on ammo and figuring out a survival strategy. The familiar pixel refresh prompt popped up, but this time, I had no way of skipping it. I could only watch in disappointment as the monitor did its thing while Jensen lost the boss fight.
The whole pixel refresh didn't take more than a few minutes, but it was enough to end my mission in the most unceremonious fashion. After that, I made sure not to take my monitor's pixel refresh pleas lightly.
Why you shouldn't ignore OLED protection
It's for your own good

Pixel refresh is one of the many ways an OLED monitor protects itself against burn-in, i.e, uneven wear of the pixels. OLED burn-in is an inevitable reality of the technology, but using pixel refresh and pixel cleaning can delay it to the point where you might replace the monitor anyway. While pixel refresh adjusts the voltage of individual pixels every 4 hours or so, pixel cleaning refers to a longer, more intensive process that kicks in after around 1000 hours of usage.
Both of these processes are essential to avoiding image retention, i.e., burn-in, on your OLED monitor, and one should never skip them. This is why OLED monitor manufacturers program regular pixel refresh cycles, so users can maintain their panels in the best way possible. You can technically skip the pixel refresh prompts a few times, but after that, you'll be forced to run it anyway. OLED monitors come with burn-in warranties, but you, as a user, play an important part in ensuring a long lifespan for your monitor.

Related
10 simple ways to prevent burn-in on your OLED monitor
Burn-in on OLED monitors is unfortunately all too real. Here are some quick tips to help you delay the inevitable.
Incorporate pixel refresh into your schedule
It's not that hard

After encountering the folly of ignoring pixel refresh cycles first-hand, I can tell you that the best way to deal with them is to blend them seamlessly into your schedule. The impeccable image quality and performance of OLED monitors go hand-in-hand with the hassle of regular maintenance, so the sooner you accept it, the better. What you can easily do is take a 5 or 10-minute break whenever your monitor wants to run its scheduled pixel refresh cycle.
It'll not only be the ideal way to take care of your expensive OLED monitor, but also help your eyes and back relax. I make it a point to get up from my desk, stretch a bit, stare outside my balcony windows to relax my eyes, and refill my water bottle if it's empty. If you have an OLED monitor, you can combine these kinds of small breaks with one longer break every 4 hours.

Screen Size 26.5"
Display Technology OLED
HDR Yes
Resolution 2560x1440
Aspect Ratio 16:9
Refresh Rate 240Hz
The Asus ROG Swift OLED PG27AQDM is still a phenomenal HDR monitor with a great OLED panel. It has a sharp resolution and a super-fast 240Hz refresh rate.
Skipping pixel refresh on your OLED doesn't end well
I know it can be annoying to run pixel refresh cycles on your OLED monitor every 4 hours or so, but it's necessary to extend the lifespan of your OLED panel. Besides, delaying it will only make it crop up at an unfortunate time, interrupting your gaming session or a crucial work meeting. Incorporating pixel refresh cycles into your schedule is a great way to make peace with OLED maintenance. A few minutes of downtime every 4 hours or so won't hurt your productivity.