In the Day of the Doctor, both 10 and 11 remember destroying Gallifrey. Does this mean that in their past, the Interface never showed up? or did he just not listen to her and destroyed the planet right away?
Hurt's Doctor is a past version of 10 and 11, so all events that happened to Hurt's Doctor happened to 10 and 11 also.
If you recall, at the end of TDOTD, Hurt's Doctor says how he won't remember any of the events that took place, and instead he would only remember destroying Gallifrey in lieu of saving it. But they don't actually destroy Galiffrey do they? So how can 10 and 11 remember destroying Gallifrey? My guess is that whenever 10 and 11 go on about how they destroyed Gallifrey, they weren't being literal in the sense that they actually recall seeing the planet explode.
The War Doctor already had the intention of destroying Gallifrey before the events of the Day of the Doctor, so when he forgot everything that took place from when he stole the Moment up until his regeneration into Eccleston, I guess that afterwards he could logically reason that he must have destroyed Gallifrey and his memory must have been wiped during the ordeal somehow. And as Gallifrey wasn't there anymore when the War Doctor regenerated, the only explanation to that would be that he destroyed it.
Also, we know 10 and 11 couldn't have destroyed Gallifrey, because the Silence wanted to kill him to prevent him from bringing back the Time Lords into the universe (and this was before the Day of the Doctor, before the 11th Doctor had the chance to see his old self again). No Time Lords== No Silence and most of the 11th Doctor events wouldn't have happened.
Actually the answer is quite simple... from what I recall the Black Archive had the memory wiping devices... At the end of the episode of The Day of The Doctor I believe there was some hints to using them to forget the events that happened that day. I believe Hurt said something along the lines of how he will forget what happened and that he will have to live with the fact that he destroyed Gallifrey, because he will be under the assumption that it was destroyed.
Also we never knew about The Silence's true role about bringing back the Time Lords until the Time of The Doctor when Tasha Lem revealed that the splinter faction of the Papal Mainframe went back along The Doctor's Timeline to prevent him from answering the question.
So there's two possibilities in my eyes, either
A. The events that occurred were always meant to happen and Hurt's Doctor and Tennant forgot the events from the memory wiping devices or because of "timey wimey" (you know how the doctor's aren't supposed to cross their own timeline)
B. Changing the past / future has major consequences which is why they had to wipe both Hurt and Tennant's memories of the events so that everything would work out as it did in The Day of the Doctor.
He was aware that the Time Lords became just as cold and unfeeling as the Daleks during the Time War. In the Time War, both the Daleks and Time Lords wiped out entire planets and civilizations, made them as if they never existed, as both sides constantly used time travel to rewrite history. The doctor could not take such widespread death and destruction, so when he had a chance to "bottle up" both sides in a far flung dimension or time stream, he took it. He feels great guilt for betraying his kind, and causing the seeming deaths of Daleks and time lords, but he did it to save the lesser civilizations caught up in the Time War.
When the newest Doctors decided not to destroy Gallifrey they created an alternate timeline and essentially another dimension where in one Gallifrey was destroyed by the Doctor and the other it was saved. Since the past Doctors and their memories were not part of the second timeline they will no memory of Gallifrey being saved.