THE HORROR
For the next few weeks, the internet will be full of weeping, wailing and gnashing of teeth. Angsty Tumblrs will be set up, proclaiming that once Matt Smith goes, they might as well cancel Doctor Who.
Certain long-time fans may start ringing the doom bell claiming that because of Name of the Doctor, Matt's replacement must therefore be THE FINAL DOCTOR EVER (because a line of dialogue from The Deadly Assassin in the 70s is reason enough to cancel the show). Remember this, because it'll be important in about 3-4 years time when the regeneration limit is ignored in the same way that UNIT dating is.
THE HATE
A new actor will be announced. They could be white, black, green, man, woman, space penguin or any combination of the above. Fandom will react as if the BBC had announced that Doctor 12 was being played by a robotic Jimmy Savile being piloted by Hitler's brain. They will froth. They will growl. They will make nasty comments about the new actor's ears.
Cue weeks of people poring over the new guy/gal/penguin's previous career in an attempt to find any reason to hate them.
Certain old fans will claim that now would be a perfect time to announce that Chris/David/Matt were all actually the Rani (it's not important) masquerading as the Doctor, and that the real Doctor is still Paul McGann, who should be re-hired forthwith. Ignore these people, as they are insane.
THE BACKLASH
By about September, we'll all be sick to death of speculation, of Matt Smith being interviewed about his reasons for leaving (he's done three years, three years is the average time for a Doctor, so expect the "Well David Tennant told me that Peter Davison explained to him that Patrick Troughton had warned him to leave after three years.") and the backlash to the Smith Era will begin. Mis-steps like Night Terrors will be thrown onto giant bonfires, and we'll all start pretending that we hated The Pandorica Opens.
If the 50th Anniversary episode isn't the best thing since sliced bread, expect the backlash to intensify and people to say things like "the only good thing that can happen at Christmas is that Matt regenerates before the opening credits." (Note. This actually popped up in a review for Planet of the Dead, the first episode after David Tennant handed in his notice.)
One or two old fans will claim that the show should be cancelled, and we should all just watch DVDs of "Time and The Rani" until the heat death of the universe. WARNING. DO NOT EVER BUY A DVD OF "TIME AND THE RANI." IF YOU FIND ONE, BURN IT. DO NOT WATCH IT.
THE ACCEPTANCE
Christmas will come, Matt will bid us adieu, we'll all remember why we loved him so, before instantly forgetting everything about him as Doctor number 12 blows us away in what amounts to a glorified end-of-episode cameo.
Those angsty Tumblr users from before will create one GIF for every frame that the new Doctor is in, before frotting themselves stupid.
Someone on Outpost Gallifrey will claim not to have watched the regeneration episode anyway, because they prefer to just listen to that Big Finish audio play where Colin Baker has to defeat an impostor pretending to be the real Doctor. They will brand the episode as "disappointing" regardless, and award it 2/10 in the voting thread.
The important thing to remember is that this has happened before. Sometimes we've had great Doctors (Pat Troughton, Tom Baker, David Tennant), sometimes we've had fairly average or downright divisive Doctors (Peter Davison, Sylvester McCoy, Christopher Eccleston), but only once have we been lumped with Colin fucking Baker, and those dark days are long behind us. So enjoy the ride, and don't get too upset.
10 times out of 11, it all turns out OK.
