Final Fantasy still warns me to "Hold RESET while you turn POWER off!"
Do I have to do this on the NES classic edition?
The NES classic is a single board computer that is emulating a NES and does not need to go into a low power state.
Nintendo presents the titles in their original format. Fortunately, it's not on the original hardware and no longer requires these certain steps to ensure data capture.
Although you can hold reset before turning the power off there is no need to. Your game is saved once that message pops up. However, hitting reset on the mini has something going for it that the original NES didn't. By hitting reset, you actually create a temporary save at the exact same location in the game as when you left it.
This is important for Final Fantasy 1, because enemy encounters are always in a set path -- by resetting you restart this path. So if you want to run the full game without ever stopping, but you need to stop because sleep/food/just a break, you can keep the same game going by using the temp save feature. When you return, you can fight your next battle as though you'd never left.