If someone from the royal family was to inherit the Founding Titan, then the predecessors' and future successors' memories would transfer to him/her as well.
But,
Eren's father steals the Founding Titan and then passes it down to Eren.
So, are the memories gone? Or will they come back if a true member of the royal family
retrieves the Founding Titan back?
Spoilers from up to Chapter 107.
According to the wiki,
only those with royal blood—the Fritz or Reiss royal families—are able to use the true power of the Founding Titan. However, if the Founding Titan is inherited by someone outside of the royal family, the power can still be used if the inheritor is in physical contact with a Titan with royal blood. This is demonstrated when Eren Yeager temporarily used the Founder's power after striking the hand of Dina Fritz's Pure Titan. It is implied by Zeke Yeager that it does not matter if the Titan of royal blood is a Pure Titan or one of the Nine Titans, such as the Beast Titan.
If the inheritor touches a human of royal blood, although the full power of the Founding Titan is still locked, some snippets of the previous inheritors' memories can be realized. The touch of Historia Reiss and her father Rod occasionally resurfaced the memories of Grisha Yeager. However, these memories are erratic and do not always come.
So if
a true member of the royal family regains the Founding Titan, I assume that they will be able to see all of the previous memories again. Naturally, they will be subject to the First King's Will.
No matter how many "others" it passes through, the original memories will remain This is at least what is implied if we are to believe the royal families current understanding of titan-inherited memories. They view the "will of the original" as a sort of curse they can't escape from if they eat the key titan and the prospect of it being temporarily in Eren and his father did not change this mindset so we can only assume that the memories would still be there.
EDIT: This quote is just the most recent example (chapter 115) where they state that if the founding titan were to fall back into the hands of someone with royal blood then the memories IE. THE VOW would come back in effect. When I originally answered this was just the royal families understanding, now with later story arcs its pretty much everyones understanding.
