Yes, they consume battery power. But battery consumption is very low. As live wallpapers depend on your phone’s CPU and GPU for rendering purposes they will consume battery.
Live wallpapers can consume more power compared to static wallpapers on your device. This is because live wallpapers are dynamic and often involve continuous animations or interactive elements, which require additional processing power and resources from your device's hardware.
The extent of power consumption can vary depending on the complexity of the live wallpaper and the capabilities of your device. High-quality live wallpapers with complex animations may have a more noticeable impact on battery life compared to simpler ones.