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.
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 Anticipator: Bond and company dive behind a dune at Crab Key as a boat crewed by Dr No's henchmen motors round the bay. One of the men shouts out, 'Come on out. We know you're there. We've been expecting you' to Bond.
Though it's implied he's not sure, and is just trying to bluff anyone there into coming out.
Anti-Hero: Bond ruthlessly executes Dent even though his gun is empty. No such event occurs in the original novel, nor in any of Fleming's novels. Word of God is that the killing was added to the film strictly to illustrate the concept of "licence to kill" and that it doesn't just mean shooting in self-defence.
Crushing Handshake: Averted—"Forgive me for not... shaking hands ..." He later demonstrates his mechanical hands' impressive strength by crushing a gold statuette.
Cryptic Background Reference: M's line about Bond replacing his Beretta was a reference to the novel From Russia with Love (and the whole scene is part of a larger portion explaining why Bond is given an "easy task" in the novel, where the Strangways incident goes without attention for weeks before they act on it). Of course, its film adaptation wasn't released yet, rendering it referenceless.
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.
What Could Have Been: The original scene would've involved hungry crabs attacking her, but it was cut as they couldn't get them to be menacing enough...which, interestingly enough, was exactly what saves her in the novel.
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.
The alleged 6-shot Smith & Wesson Dent uses (which correctly should have been a revolver) is actually a suppressed 7+1 Colt 1911 automatic, the slide of which locks back after the sixth shot anyway, and then returns to battery on its own after it initially leaves Dent's hands. Additionally, Smith and Wesson didn't manufacture a 1911 at the time anyway.
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.
A subversion with Professor Dent. Bond asks him who he's working for, and Dent replies: "Well, you might as well know as you won't live to use the information. I'm working for-" before grabbing his gun and attempting to shoot Bond. Sadly for him, he's had his six.
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, even though Bond states it to the receptionists in Dr. No's base (It's hidden in the narrative). 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.
Right Hand Versus Left Hand: Quarrel initially assumes Bond is just a nosey enemy, and leads him into an ambush when Bond tries to find out more about what happens to Strangways. Leiter gets in on it before all the misconceptions are sorted out.
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, although Dr. No was at least attempting to be discreet, as he also sends someone to poison a basket of fruit delivered to the same room earlier on.
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.
You Have 48 Hours: Leiter tells Bond the investigation has to go ahead before the next space programme launch, but it's never a major factor in the plot other than getting Bond and Quarrel to Crab Key.