Adaptational Modesty: Occurs during the iconic scene where Honey Ryder comes up out of the sea and walks up the beach. As generations of short-changed Bond fans have pointed out, in the novel, Honey Ryder is naked when she emerges from the ocean. In the film, Ursula Andress wore a white bikini that has become associated with her character and the idea of the Bond girl as a symbol of glamor, sophistication, sex appeal, and danger.
Air-Vent Passageway: Double-subverted. When Bond tries to escape his cell through the vent, he gets shocked when he touches the grill. However, he tries again by using his shoe to push it out and succeeds in escaping. As a nice touch, he experimentally taps the grill at the other end with his feet to make sure it isn't electrified.
Also justified in the novel. It's designed to be a part of a Deadly Game.
And the reason it's so wide? It's not an air vent; it's a water vent, as Bond learns to his dismay.
Animal Assassin: The poisonous spider (in the film) or centipede (in the book) which gets dropped into Bond's room, as well as the one Honey put in her landlord's bed after he raped her in the book (in the film, she says she uses a black widow spider).
The book's Deadly Game also had a section with scorpions; upon reaching the end, Bond did battle with a giant effin' squid.
Subverted in the book - she was tied up with the threat of being eaten by a swarm of crabs. She knew the crabs were harmless, and let them swarm over her; her distressed attitude was over what would happen to Bond.
Discontinuity Nod: M's line about Bond replacing his Beretta was a reference to From Russia With Love in the novels. Of course, From Russia With Love wasn't released yet, rendering it referenceless.
Bond does possess a glow-in-the-dark watch; these were relatively uncommon at the time, as the only glow-in-the-dark pigment then available involved radium, which is both expensive (accounting for rarity) and radioactive (accounting for its ability to trigger a Geiger counter).
Q is described as being the armorer, so technically the film's main gadget was the Walther PPK Q gives Bond. While old hat, it's worth noting that in films of the day people were used to seeing revolver-style handguns, so a "modern" semi-automatic magazine-fed handgun would probably have seemed rather futuristic when the movie was made.
Bond's killing of Dent qualifies. No such scene occurs in any of Fleming's novels, but because the film version of Bond was promoted as a character who "kills who he pleases, when he pleases, how he pleases", and the novel actually has very little in the way of Bond killing people, this scene, along with a later sequence in which Bond knifes a guard for no real reason other than to have Bond give some justification in dialogue for his actions when Honey acts shocked, were added. Connery's Bond never acted this way again, though Moore and Dalton had a few Pay Evil unto Evil moments, and it's become normal operating procedure for the Craig version.
What Could Have Been: In the film Bond shoots Dent off the bed, then delivers a second shot to the back of the still-breathing bad guy. As originally filmed, Bond actually emptied his entire magazine into Dent (echoing the "You've had your six" line), but this was cut as being too violent. Even today, how many shots Bond fires on screen (one or two) depends on who is broadcasting the film.
Guns Do Not Work That Way: A pair of scenes are taken almost verbatim from the novel, but with the guns replaced by ones that make no sense for the scenes in question.
The 6-shot Smith & Wesson Dent uses is actually a suppressed 7+1 Colt 1911 automatic, the slide of which locks back after those six shots anyway, and then returns to battery on its own after it initially leaves Dent's hands.
The Beretta M forces Bond to surrender is the M1934 in .380 ACP. In the books, Bond carried a 418 in .25 ACP, which got caught on his holster when he attached a suppressor, allowing his enemy to shoot him (here, it's stated to have jammed). It's replaced by a Walther PP, in the same caliber, but is stated to be the shorter PPK in 7.65mm, which would in fact be inferior to the M1934.
Handy Cuffs: When Bond is captured by the crew of Dr. No's "dragon".
Just Between You and Me: A not too blatant example, since the US had worked out before the events of the film that their rockets were being toppled; they just didn't know who the culprit was, and Bond works out by himself that Dr. No is responsible. However, Dr. No also freely gives away the existence of SPECTRE, who Bond and, presumably, MI6 had been totally ignorant of until that point. In fairness, he only told him because he was trying to recruit him, and Russia and China definitely know because they keep doing business with them (No is on a mission for SPECTRE, but it is strongly implied that they were hired by Red China; No even has an army of Chinese henchmen), so its not so bad if half the world knows anyway.
The Load: Honey is as useful as a rubber spanner. About all she does is help find the "dragon" that takes them to Dr. No's base. We know she's there for the Fanservice, to play the Distressed Damsel and get rescued. And for the bedroom cut-away as the credits roll.
Mauve Shirt: Quarrel, ironically wearing a Red Shirt. Killed by a flamethrower tank painted to look like a dragon.
No Name Given: The three black assassins are only known as "Three Blind Mice" after the song that plays in their introduction.
M's name is never revealed. Indeed, although Fleming would do so in his books, the original M's full name would never be revealed in the films (his first name, Miles, would be uttered in The Spy Who Loved Me); not until Skyfall would an M's complete name be revealed.
Red Right Hand: Dr. No's mechanical metal hands. Although they're moderately maneuverable and super strong in the film, they're little more than crude pincers in the novel.
Their explanation differs between book and film. In the book, his hands were cut off by the Tong; in the film, they were damaged in his radiation experiments.
Reed Snorkel: Used by Bond, Quarrel and Honey Rider to avoid Dr. No's guards.
Same Language Dub: Ursula Andress has a very thick accent, so she was dubbed not once but actually twice in the movie. Nikki van der Zyl dubbed all of her dialogue, while the calypso was sung by Diana Coupland. Indeed, van der Zyl dubbed the voices of several of the women, including the original Bond girl, Sylvia Trench; the secretary to Strangways, Mary Trueblood; and Kodak fiend Annabelle Chung.
You can hear the original voice of Eunice Gayson (Sylvia) in one of the original trailers for the film.
Sleeping Dummy: Bond uses several pillows under the covers of his bed.
Spiders Are Scary: A mook plants a tarantula in Bond's hotel room at night, clearly the most terrifying thing in the world judging by the shrill soundtrack music and the obvious pane of glass between the spider and Sean Connery. In the book it was a centipede known by Bond to be deadly - guess he hadn't read up on all the arthropods...
As Cracked pointed out, it would have been more effective to put a guy in there. With a gun.
What the Hell, Hero? A bit of irony: Bond does this with Honey when she describes how she murdered a man who raped her by putting a black widow spider in his bed, causing him to die over the course of a week. "I wouldn't make a habit of it" he says, shocked. But later, after Bond somewhat unnecessarily stabs one of Dr. No's men to death, Honey acts shocked and asks why he had to kill the man.
Viewers used to the more light-hearted Bond films (pre-Daniel Craig) often have this reaction to the Dent shooting scene.
You Have Failed Me: Subverted; Dr. No deals with Professor Dent's failure to kill Bond simply by having him resort to an Animal Assassin and hope for the best. When that fails, he's lucky to be captured and killed by Bond during an attempt to deal with him personally.