Recently some exoplanets were discovered which are thought to be the leftover cores of giant planets that had survived a star becoming a Red Giant and then shrinking back to a subdwarf.
If giant planet cores can last the lifetime of a Red Giant, then maybe smaller planets can survive at least for a while.
When the Sun becomes Red Giant, if the Sun engulfs the Earth and Moon before they have evaporated, what will happen to the Earth-Moon orbit ? Will they be crashed into each other, or will they drift apart or will the orbit remain stable for a while before they are melted and evaporated away?
Edit: Newly found planets are 'roasted remains' BBC News, 21 December 2011
Edit: More planetary systems around white dwarfs (WD 0421+162 and WD 0431+126) that shows signs of planets being consumed by the stars. http://news.discovery.com/space/alien-life-exoplanets/hubble-discovers-planet-graveyard-130509.htm