Screen stays on while charging... how do I fix this?

A couple of months ago I had to get my phone replaced with a refurbished one. Since then, I have noticed that my replacement phone behaves differently with regard to the display while charging, and I can't figure out how to change the settings to make it like my old phone.

With my old phone, when I plugged it in to charge, the screen would turn on for a few seconds then turn off. Likewise, after becoming fully charged or after receiving a notification, the screen would light up for a few seconds and then go dark again.

With my replacement phone, however, when I plug it in, the screen turns on and stays on, unless I press the power button to turn the screen off. And then, if I receive a notification or the phone becomes fully charged, the screen will turn on and stay on until I manually turn it off or unplug the phone. This bothers me because I imagine the screen must be on for most of the night, and I wonder if this could possibly burn an image onto the screen or cause other issues.

Anyone know how to fix this?