When Lionel wonders why he should take Albert seriously.
Albert: Because I have a voice!
Lionel: Yes, you do.
The end credits, which mention that George VI rewarded Lionel's services by making him a Commander of the Royal Victorian Order — a special order of knighthood reserved for personal services to the monarch.
Poor Lionel, though. Only the top two grades of the Royal Victorian Order (Knight/Dame Grand Cross and Knight/Dame Commander) give the recipient the title of "Sir" (or "Dame"). Lionel's CVO was level three - one grade below knighthood.
He did say that George VI's physicians were idiots and when told they were knighted he responded "That makes it official then!" so I don't think he cared. (In fact, if this exchange actually took place, this may bewhyhe was given the CVO, especially considering that, as wonderful a man as he was, Bertie would have been a less effective rallying standard for the country without Logue's help.)
A CVO actually has more significance in this case. A knighthood could be given to award anyone but a CVO reflects personal service to the monarch meaning that what Logue did for the King was worthy of national recognition.
Near the beginning of the film, Bertie telling a story to his daughters, despite his stammer. Both the moment and the story he tells are incredibly adorable.
Made even cuter when he hugs them at the end of the story.
The best part of all: Colin Firth himself came up with that story, which he told to his own kids.
The scene when George VI arrives at Lionel's home to apologize for his behavior, along with the Call Back to earlier in the film.
George VI telling the archbishop he wants Lionel to sit in the King's Box for the coronation.
Archbishop: But your majesty's family sits there!
George VI: That is why it is suitable.
Pretty much the entirety of the Westminster Abbey scene is a CMOH when it's not being awesome.
Rather understated, but I found the moment where Winston Churchill encourages King George VI right before his big speech by informing him that he himself (i.e., one of the greatest orators of the twentieth century) once had a similar speech impediment to be very touching.
Elizabeth's comforting of Bertie as he cries over his accession. "But then I thought, 'He stammers so beautifully, they'll leave us alone.'" It's in that moment the entire audience sees just how deeply she loves her husband, and what she gave up to be with him. Lady Elizabeth Bowes-Lyon no more wanted to be Queen than her husband wanted to be King. She would go on to be one of the most beloved members of the royal family - at times, even more so than her daughter the Queen Regnant - for over seventy years.
And right after Bertie finishes his speech, she gives him a little kiss and tells him "I knew you'd be great." Awwww...
After the Speech, George VI gives a hug to his daughter, the future Elizabeth II. This is especially poignant when reflected against the line from The Queen by Tony Blair about her "taking a job she watched kill her father." Truth in Television, of course: Elizabeth was very close to her father in real life.
As a bonus, the real Queen Elizabeth adored this film for the warm portrayal of her late father.
Right before Bertie gives the speech at the end.
Lionel: Forget everything else and say it to me. Say it to me as a friend.
Colin Firth and Geoffrey Rush were given level billing before the title — sweet, considering the relationship between the men they played.
Right before he gives the climax speech, King George VI tells Lionel that no matter what happens, he's grateful for all he's done for him.
As much of Jerkass as he is in the movie, David/King Edward's graceful abdication is this trope. He doesn't belittle his brother, rather he swears his allegiance to him, unwaveringly.
One might notice that Lionel stops conducting Bertie during the last third of his speech. He knows he's got it under control now.
Last but not least, the final line of the postscript:
Lionel and Bertie remained friends for the rest of their lives.