Fridge Horror: Tatiana, a loyal Soviet clerk that's good at her job and has no malice against anyone, is manipulated by a traitor through her devotion to country to commit treason and played by Bond to defect out of infatuation as the audience fully knows Bond has no intention of settling down with her, likely leaving her to an uncertain future in an enemy country.
Karim's off-handed remark that the West and the USSR's respective uses of Gypsies and Bulgars as muscle has ignited a blood feud between them.
Fridge Logic: Couldn't the Gypsy Prince decide FOR HIMSELF which girl to marry?
In this film Bond faces off against what would be the first in a long line of burly blonde Aryan looking henchmen in the form of Donald "Red" Grant. A little less than 50 years after this film was made the world was given a blonde James Bond (in contrast to the usual dark haired Bonds) in Daniel Craig and darn if they don't look similar.
Speaking of which, Connery and Shaw would have a rematch 13 years later as Robin Hood and the Sheriff of Nottingham in Robin and Marian.
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..
Narm Charm: Having epic music for that mundane scene arguably makes it better. Also kind of helps that Bond kills someone (Bond One-Liner and all) and the music kicks in right on the scene before that.
Bond being chased by a SPECTRE helicopter (not unlike Cary Grant in North By Northwest) and taking it down with his portable sniper rifle.
Special Effect Failure: The actor playing Kerim Bey dabs his arm with a red sponge to simulate being shot during the gypsy camp fight.
Tear Jerker: In-Universe, the death of Karim Bey (which also pisses off Bond). Meta: Pedro Amerdantriz (Karim's actor) was dying of cancer during filming, and on various scenes he couldn't walk (the running Bey, Bond and Tatiana do through the Istambul market was done with a body double for Karim).
Complete Monster: Donovan "Red" Grantnote changed to Donald in the film 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. Bond's chosen executioner, Grant kills his ally Darko Kerim and proceeds to try to murder Bond and his lover, admitting that all he cares about in life is his ability to murder people.
Funny Moments: Tatiana asks why British men don't wear cologne like Russian men do. Bond replies "We wash."
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.