Turning off animations will save a bit on battery, not as much as turning off tactile feedback, dimming the screen, and setting longer refresh rates for your apps, but yes it will save a tiny bit on battery. The down side to this though is that depending on your OS you may loose some functionality. In some versions of Android turning off animations prevents menus from opening. This was the deal breaker for many on either KitKat or Lollipop, I can't remember which, and I'm not 100% that it was ever fixed in those or preceding versions.