One of the potent themes of Star Wars: The Force Awakens is the weight of a newly inherited legacy. The Force Awakens was effectively a passing of the torch: Both the relatively unknown actors in real life and the characters they were playing on-screen were entrusted with the Star Wars legacy from the the older cast. Now, narratively, this inheritance includes Rogue One, another film that has left a ripple effect in the Star Wars universe. So, how could Rogue One's story affect Episode 8, the next film in the Star Wars saga that will return to the shared adventures of Rey, Finn, and Poe Dameron?
Well, all I can safely confirm about Episode VIII is that it is indeed a Star Wars film, it was directed by Rian Johnson, and it features the newest aforementioned triumvirate of heroes. Rey, Finn, and Poe have all met and interacted with Leia and Han; Rey is the only new character who's met Leia's brother, Luke. There are references to the battles Leia, Han, and Luke fought against the Empire. Throughout the course of The Force Awakens, Rey and Finn become aware of the true burden of those battles and the ripple effects of the Empire's rise and fall. We also have the connection between Darth Vader and his protégé, his grandson Kylo Ren (Kylo talks to Darth's rusting helmet in The Force Awakens).
Interestingly, though, there is barely any mention of what happened when Leia received the plans from what we now know to be the Rogue One team during The Force Awakens. Yet it was precisely because of the actions of the Rogue One team that any events occurring after their final battle on the planet Scarif would not have occurred; I'm sure a much darker Star Wars timeline would have happened.
Thus, the most immediate effect that Episode VIII would be affected by Rogue One would simply be Rey, Finn, and Poe learning of the mission at the heart of Rogue One, either individually or separately. A name drop or quick history lesson could definitely happen depending on how much time Rey spends in Luke's company, especially, but also whether or not Leia would return and interact with Finn and Poe. It would make sense — and be in line with the theme of legacy — for Rey, Finn, and Poe to learn of what happened to the Rebellion before they came along if only for them to learn not to repeat the mistakes of the past.
Speaking of Rey, there are some who believe there may be clues in Rogue One that point to her parents, particularly with the film's heavy focus on Mon Mothma, who many note has the requisite Coruscanti accent in which Rey also speaks.
These are a few of the safest conjectures on the ripple effects of Rogue One through to Episode VIII. Without so much as a concrete teaser from Episode VIII to go on, it's tough to draw any further conclusions. But the release of Rogue One before Episode VIII was crucial to appreciating the original Star Wars trilogy and the upcoming Star Wars films.
The world of Star Wars is, in reality and in its story, so steeped in its own rich history that it's hard to ignore the weight of it as time goes on. I don't doubt the effects of Rogue One will be felt and commented on in some capacity in Episode VIII, especially now that the former's release gives the latter an ability to do so without spoiling it for the fans.