In "The Five Doctors", the Doctors remember their previous encounters with each other. Two also remembers Omega just fine. And he knows that Jamie and Zoe had their minds wiped, even though that happened just before he turned into Three, so there's no way for him to be aware of that when he's just travelling about freely. Don't think about it too hard.
Of course, if you're obsessive enough, you can explain away that retcon by constructing a whole new season of stories, causing all sorts of new problems with other continuity. Needless to say, fans have already done this.
When the 10th and 5th Doctors meet up during the Children in Need Special "Time Crash", the 10th is in shocked disbelief to be seeing his former self, then goes on to use memories he picked up as the 5th meeting his future self to defuse the situation. When the illogic of this is brought up (not to mention the violation of multi-doctor meet up Canon established from the other 3 times this has happened), both Doctors mumble something about "Timey Wimey" and move on.
It's actually mentioned in the old series of Doctor Who that the Time Lords deliberately took Gallifrey out of its own time to prevent any potential rogue Time Lord from ever altering Gallifrey's history. Then again it's also stated time and time again in the old series that Gallifrey can't be destroyed, and look what they did in the NA and the new series. note The Doctor is just that good. Except all of him are able to travel into the Time War and save Gallifrey.
In "The Fires of Pompeii", a companion asks why the Doctor will thwart aliens but not stop a particular historical catastrophe, and the Doctor replies that some points in time are fixed, while others are in flux. His being a Time Lord allows him to perceive which is which, and act accordingly; even against his nobler instincts. It's revealed in the climax that the reason he can't change the catastrophe is because he's the one responsible for making it happen.
The Doctor has also stated in the past that his knowledge of history is "perfect"; this may mean that he knows exactly what he may or may not change.
And if there's any discrepancy, it's obviously history that got it wrong!
More about Fixed points in time in "Sarah Jane Adventures". Sarah Jane's parents dying is a fixed point in time because she has seen it, remembers it, and knows it happened. Changing that would be bad.
The Doctor tried to mess with a fixed point in "The Waters of Mars." It doesn't end well. He explicitly states that there are fixed points in history which cannot be changed. Those points in history greatly affect the future and allow for time to follow a more or less consistent path. Anything he does to try and change history will simply cause the event to occur regardless. Even the Daleks are shown to respect this. The Doctor, feeling frisky, tries to alter one. Events remind him that even a Time Lord has limits.
This is even further confused, though, because at the time the Dalek respected the fixed point by not killing a child who would grow to be important in human history, they were working on a plan to destroy reality. So you can't alter a fixed point, except the Daleks were going to by destroying her along with all of reality.
This event could be explained by the fact that destroying reality would distort events to the point that the fixed points that applied in the original history would no longer apply, whereas killing the child would be denying a fixed point while it still applied to history, which would be bad, but that's getting into WMG territory.
Of course, the whole "the Dalek was respecting a fixed point in time" thing was just the Doctor's speculation. It's just as easy to assume that the Dalek left the child alone because, at that moment, it was recalled to Davros's flagship in preparation for the detonation of the reality bomb.
In "The Wedding of River Song" we finally get to see what happens if you alter a fixed point too much. All of time collapses, happening at once. You'll have Holy Roman Emperor Winston Churchill riding around on his personal mammoth while they discuss the political pressures caused by the War of the Roses, greet a Roman Centurion, and see a Silurian doctor for a check up. Meanwhile dinosaurs are in the park, and Charles Dickens appears on the news to talk about a book he'll be releasing at Christmas. Only some people will be able to hang onto their memories of 'correct' time. The date and the time will never ever change. They are the date and the time of when the fixed point was supposed to happen. If allowed to continue, time itself will break, causing THE DESTRUCTION OF REALITY ITSELF!.
The Doctor Who New Adventures had the concept that Time itself was a sentient entity who consciously fixed various timeline hiccups resulting from time travel with the Doctor as her champion.
In the first season finale of the NS, the Doctor says that the TARDIS protects itself from paradox. Whenever and wherever the TARDIS lands, the events that led it to go there, and led to the world it's in once it's there, become unalterable.
In "The End of Time" the Doctor attempted to explain a Time Lock to Wilfred.
Doctor: They're sealed inside of a bubble. It's not a bubble, but just think of a bubble.
The Trope Naming episode, Blink, actually involves a mostly-internally-consistent Stable Time Loop. It's the show as a whole that fulfills the trope by being inconsistent.
The whole of The Big Bang is built on this trope — The Doctor saving the day and escaping from the Pandorica is built on an ontological paradox — he shows up already escaped to enlist Auton!Rory in effecting his escape. The Doctor even explains that this would normally cause drastic side effects for the universe, but luckily the universe had already been destroyed.
The affair was referred to in a later episode
River: He's interacting with his own past. It could rip a hole in the Universe.
Amy: Yes, but he's done it before!
Rory: And in fairness the Universe did blow up.
A Christmas Carol also features this heavily. It starts with the Doctor showing a video Kazran made as a boy to the older him - and traveling back in time to when he made it, leaving Kazran watching a video of the Doctor interfering in his past as his own memories change to reflect that this had happened. Kazran then has memories of not growing up while being visited by the Doctor, and memories of being visited by the Doctor. He then ponders how he's never met the Doctor before tonight, but seems to have known him all his life. It ends with the Doctor showing the younger Kazran the man he turns into, leading to the older one having a change of heart partly brought on by realizing he's turned into his father, and partly by him being retroactively altered by the experience of being horrified at seeing his older self as a boy.Oof. It's implied this method is far from perfect, as Kazran's own mind-reading controls no longer recognize him, despite the fact that they should logically have been programmed for the Kazran that existed in the current timeline. I think it's that the Doctor changed the boy as well as the man. Hypothetically, Boy Kazran went through all the stuff that Scrooge Kazran did, but had the additional factor of seeing Scrooge Kazran and never wanted to become like he was. Presto Change-O and immediate echoes into the future... his brain waves change - creating a new Kazran who both experienced Abigail and Scrooge Kazran and also maybe even at one point of the new history we didn't see rejected his father (so no actual mind control for him was made). Who knows? It is a Timey-Wimey Ball after all
River Song. Her encounters with the Doctor are not synchronized at all. The journal checking seen in "Silence in the Library" and "The Impossible Astronaut", as well as the "spotter's guide" from "The Time of Angels", seem to indicate that she meets the Doctor in a random order, but when River's past/future with the Doctor is brought up in Series 6, it's implied that they're traveling in practically reverse order — the kiss at the end of "Day of the Moon" is implied to be River's last because it is the Doctor's first. Despite the fact they clearly aren't meeting in reverse order, since the Doctor meets her months after she was born four times after he 'first' meets her. And she doesn't recognize Rory in "The Big Bang", despite seeming to know him already in "The Impossible Astronaut", which is earlier in her timeline (though that was most likely due to him having been erased from history at that point). Their meetings are mostly random, and any given time the two meet up may be synchronized, but — overall — they're moving in opposite directions.
River: Rule 1. Amy: The Doctor lies. River: So do I. All the time. Have to. Spoilers.
To add to the weirdness that is time-travel in Doctor Who, look at its opinion on the Blinovitch Limitation Effect. In some cases it seems to suggest that Never the Selves Shall Meet, lest they cause reality to shatter. Or maybe that's only if there's another paradox nearby. Sometimes it causes memory loss if the two touch, like what happened to The Brigadier. Maybe the same object touching will just cause sparks. Or maybe nothing will happen at all except flirting. It's just whatever happens to work for the plot.
"Father's Day" summed it up pretty well. Pete Tyler being alive created a paradox, and anything else would make it worse. So yeah, interacting with one's past self makes sparks, and a paradox fills the air with gas fumes (sort of. Not really at all, but if that helps just think of it like that).
And just because the DW section for this trope needs to be larger, used extensively in the episode "The Girl Who Waited". The TARDIS crew happens upon the 'Two Streams' health centre. They take people who have contracted fatal illnesses, and place them in the 'fast' stream, symbolised by a red water-fall. They can live their whole life and age normally in only a day. Meanwhile, their loved ones are in the slow stream, symbolised by a green anchor, and can watch their lover/family/friend have a fruitful life. Unfortunately, it all goes wrong when Amy gets trapped in the fast stream. Eventually Rory manages to break in to save her, but 39 years have passed, leaving his wife old and bitter. He can jump back in time to save younger Amy, but can only do so with older!Amy's help. Except she doesn't want to be re-written and stop existing. Eventually they decide to save both of them by breaking the laws of causality; at the last minute the Doctor reveals this is actually a paradox and leaves Old!Amy behind to die.
Perhaps the earliest use of this in the series is "The Space Museum". The First Doctor and his companions arrive at a planet with a space museum in it, but due to the TARDIS "jumping a time track", they arrive Just One Second Out of Sync, rendering them invisible and inaudible to anyone else. While there, they see themselves trapped in museum display cases. When their Invisible Main Character status wears off, the cases go away, they're still inside the museum, and have to escape or otherwise find a way to avoid the fate they saw for themselves. The Doctor claims that time has alternatives.
"Day of the Daleks" may have been the Trope Codifier for Timey-Wimey Ball in Doctor Who. Guerrillas from an alternate 22nd century try to assassinate Sir Reginald Styles to prevent him disrupting a peace conference, which caused wars enabling the Daleks to invade Earth. In the process, they disrupt the conference themselves. However, the Doctor is able to travel back from this alternate future and stop the guerrillas.
"Pyramids of Mars" has the Doctor show Sarah Jane how time has its alternatives. Even though Sarah Jane is from 1980 and knows the world wasn't destroyed in 1911 by Sutekh, the Doctor takes her to 1980 and shows Earth has been destroyed as they didn't stop Sutekh escaping. This is partly accounted for, the Doctor says individuals can shape the future but only powerful beings like Sutekh can destroy it.
Big Finish Doctor Who was using this trope from early on. The 8th Doctor saving Charlotte Pollard from her death on the R101 causes a paradox, meaning anti-time starts infecting the Universe, causing odd things with history to happen during the 8th Doctor audio stories leading up to Zagreus. For example, Shakespeare has disappeared from history (which is explained in "Time of the Daleks") and Benjamin Franklin was President. Finally in "Neverland" Charley helps save the Web of Time, meaning that the paradox and anti-time infection become part of the Web of Time. To complicate matters further, she later travels with the 6th Doctor, even though she shouldn't be alive at his point in the Doctor's timestream.
"Seasons of Fear" has a very complicated Timey-Wimey Ball. The Doctor goes back in time to stop Sebastian Grayle, because Grayle prompted him in an artificial alternate timeline in which the Doctor hadn't even met him yet. Grayle then develops a hatred for him, eventually leading to him creating an artificial alternate timeline. To make this more complicated it isn't clear how the Doctor met this Sebastian Grayle as in 1806 Grayle goes back in time and is killed by his past self.
"Flip-Flop" features a very odd version of this trope, with an apparently Stable Time Loop between two alternate timelines, meaning there are two 7th Doctor and Mel(s).
"Jubilee" involves the Doctor going into a divergent universe but about a hundred years after it has diverged, with the Sixth Doctor experiencing Flash Sideways and remembering being in a Dalek war a hundred years ago. Then the Doctor ends up accidentally causing the war a hundred years ago in the past that created that divergent timeline, only it's actually happening then, as well, because the Doctor's presence caused the timelines to merge, somehow. Then the Dalek survivor of the war a century ago talks the invasion fleet into suicide, which unmakes the alternate universe and resets the timeline. It makes a lot more emotional sense than it makes logical sense.
In "The Day of the Doctor", the Eleventh Doctor is brought by the Moment to meet the Tenth Doctor and the War Doctor (an incarnation between the Eighth and the Ninth Doctors, whom the others don't recognize as a Doctor due to... questionable actions). The War Doctor gets annoyed at the childish things his future selves say, including "timey-wimey" (as stated by the Eleventh Doctor). The embarrassed Tenth Doctor (who invented the phrase) says he has no idea where the Eleventh Doctor picked this up.