Do screens still need to be saved?

OLEDs are famous for burning in[1], so potentially "yes".

1: https://www.pcworld.com/article/2918628/your-oled-displays-w...

Don't tell me they burn when showing nothing or going to sleep?

We used to have screensavers with CRT because it took them a lot of time to power on and most early CRTs didn't have power saving capabilities so showing something wasn't significantly worse than having a black screen in term of power usage.