Big-Lipped Alligator Moment: The really long Cat Fight between two skimpily-dressed Roma women has nothing to do with the overall plot and is pretty Narmy in its attempts to be titillating. Toned down from the book, where both women end up completely naked.
As far as Bond's one-shot allies, Kerim Bey is one of the most memorable and charismatic. His actor Pedro Armendariz is also something of a Memetic Badass for delivering such a bombastic performance while dying of cancer.
Kerim Bey, watching over a bound and gagged criminal, with hours to kill: I have had a particularly interesting life. Would you like to hear about it?
Bond helps a Soviet defector get away with a top-secret piece of Soviet equipment. In The Hunt for Red October, Sean Connery would star as a Soviet defector trying to get away with some top-secret Soviet equipment. At least one reviewer humorously dubbed Red October "From Russia With Sub". Also, a portrait of cosmonaut Yuri Gagarin is prominently featured inside the Soviet embassy. In Red October, Capt. Ramius (Connery) would extol Gagarin in his speech to the crew.
Moral Event Horizon: For most of the film, Grant appears to simply be a professional doing his work. Then he and Bond finally meet and we see what a twisted bastard he really is.
"My orders are to kill you and deliver the Lektor. How I do it's my business. It'll be slow and painful. The first shot won't kill you. Nor the second. Nor even the third. Not until you crawl over here and kiss my foot!"
Narm: Bond checks into his hotel room and casually walks around checking for bugs... with the Bond theme playing in the background the whole time. It's like the Bond canon version of Mike and the bots blurting out "BA-DA-BA-DAAAAAAAAAAA!" during the boring bits of Agent for H.A.R.M.
When Blofeld gives his Siamese fighting fish speech, where he explains that the third fish is letting the other two fight so that it can finish off the survivor, it is easy to see that the real reason why the third fish isn't attacking is because the fish tank is divided by a glass wall.
Pedro Armendáriz dabs his arm with a red sponge to simulate being shot during the Romani camp fight.
Bond slapping Tatiana after Kerim's murder. Granted, Bond is understandably upset over his friend's death and he knows Tatiana has been lying to him, but it seems quite brutish to modern viewers.
It's a throwaway line but Kerim casually notes that the Western allies and the Soviets in Istanbul have encouraged a blood feud between the Roma and the Bulgars through using them respectively as pawns in the Cold War.
On that note, the representation of the Romani is... Very dated. The use of the common slur term for the people rather than the more appropriate Roma, the representation of them as simultaneously filthy and backwards and the women as sexual objects of desire (not to mention the weird-as-hell catfight to the death) definitely shows how attitudes towards the Romani have changed in the last 60 years.
Complete Monster: Donovan "Red" Grant has been afflicted with a burning urge to kill during the full moon since his youth. Starting with animals, he graduated to vagrants and tramps before becoming a Serial Killer who targeted young woman in his native Ireland. Escaping justice by joining the military, Grant would defect to the Russians after he realized they could offer him the bloodshed he craved. A ruthless, efficient killer, Grant rose to the position of head executioner of the clandestine group SMERSH and was indulged in being allowed to murder with impunity during the full moon. This even extended to Grant being allowed access to prisons with chainsaws during those nights. He is chosen to execute Bond and Tatiana.
Bond finds Q Branch's attaché case (which contains two hidden knives, gold sovereigns and his silencer, but neither the tear gas nor the rifle from the film) to be a bit over-the-top (though of course he ends up using everything), and later he laments that his side doesn't provide handy gadgets like Grant's five-shot copy of War and Peace. It's pretty strange to see the iconic "spy gadget" user lamenting that he doesn't get kit like that.
Kerim's kidnapping, imprisoning, and attempting to brainwash a woman into loving him in his Backstory is portrayed as a mere "youthful indiscretion" that the Service quickly straightened out of him, rather than the serious crime it would be extremely difficult to redeem oneself from that it would be seen as today. This can definitely hamper modern readers' sympathy for Kerim.