Yes, of course. Although in most cases, the appearance of living backgrounds lures us with Its movement, diversity and mystery, we have to think twice before we download them because they are pulling a lot of energy.
Yes it certainly does, I tried both live wallpapers and static wallpapers and the difference in battery consumption is huge, If I have a static wallpaper my phone lasts up to 1 day by standard usage, but with live wallpapers it only lasts up to 4-5 hours. I mean that says a lot!