From my research burn-in is caused more by static images being constantly on at high brightness. I've been paying attention, and Apple seems to do things to mitigate this. (The clock, wifi status, etc. changes from white to black depending on app background color, along with the home bar. Keeping auto brightness on will help too. I think True Tone will help as well because whites are shifting in color so they don't remain the exact same at all times.
As far as battery life goes, I'd say that the power savings are minuscule to negligible using dark wallpapers. I think it was Jerry over on Android Central that wrote an article about it more recently. Generally people aren't on their home screens long enough to really save much battery by choosing dark wallpaper. Now if there was a true dark mode that uses black backgrounds at hex black
#000000 it would probably save some battery, but if it's not hex black the pixels will still light up.