Yes, OLED screens can get burn-in, meaning a permanent faint ghost of a static image, but it is uncommon with normal viewing. LG says burn-in and image retention are possible on virtually any display, and that most cases come from static images or on-screen elements, such as channel logos, shown uninterrupted for many hours or days, usually at peak brightness.
Short-term image retention is different: it usually fades after the TV is turned off for a while or after a few hours of varied content. Modern OLED TVs also include protections such as an automatic screen saver for static images, a screen shift feature that moves the picture slightly, logo brightness reduction, and a panel refresh that clears the pixels.
Source: LG USA, “OLED TV Reliability: Burn-In & Lifespan”
Image: LG OLED TV by LG Electronics, CC BY 2.0, via Wikimedia Commons.