There is a room designated as the President's Bedroom on what is called the Second Floor, but it is actually the third above ground floor in the White House. There are numerous other bedrooms (4 or more) on the same floor, so the president probably has some liberty to choose where he would prefer to sleep, unless it is mandated by law. However, I have seen nothing to indicate that is the case. The private quarters of the First Family may be furnished however they wish, and frequently the new president brings in his own furnishings to personalize his living space. I would imagine that the president has his own preference for the type of bed he sleeps on, so I cannot see that he would be forced to sleep on anything he considers uncomfortable. So it is unlikely that presidents sleep on the same bed from administration to administration.
To find out more, just google "White House presidential living quarters," or something similar.