Surprisingly enough, the original version of The Joker. In his first appearance, he was a permanently smiling psychotic gangster with no sense of humor whatsoever. In his first fight with Batman, Bats is actually the one making puns, while Joker is screaming "I am going to kill you!"
The Joker was actually able to make Batman laugh at the end of the graphic novel The Killing Joke (even though the story as a whole wasn't funny in the least).
X-Men: Professor X and Magneto are frequently flanderized as no-nonsense extremists. Patrick Stewart's performance is the first attempt to move away from this, although one might simply assume that as he gets older, he will get more serious. Also, Ian Mc Kellen gave Magneto a deadpan, cynical sense of humor which was expanded on by Michael Fassbender.
Film — Live-Action
The Film Rat Race features a character who tragically was born without a personality
In No Country for Old Men, according to Carson Wells the big problem with Anton Chighurh isn't that he is batshit insane, but that he lacks a sense of humor.
The various Terminators exhibit this behavior, being emotionless robots.
From Happy Go Lucky, Scott the driving instructor, and Poppy's older sister Helen.
In Highlander, The Kurgan accuses nuns of being this.
Agent K in Men in Black quips (deadpan) that "we at the FBI do not have a sense of humor we are aware of."
This, and attempts to develop a sense of humor, are the focus of Sha Wujing's character arc in The Lost Empire (a sort of unofficial sequel to Journey to the West.) He's eventually told that it's better to have no sense of humor than to have the sense of humor that the villains have, and he laughs for the first time at the realization that he was worried over nothing.
In Going Postal, Adora Belle Dearheart tells Moist Von Lipwig that she has no sense of humour due to her embarrassing name.
In Making Money, the head bank clerk, Mavolio Bent, has "no sense of humour whatsoever. It has been proven by phrenology." It becomes more apparent when one considers his background as the child of a clown, being laughed out of the arena during his first performance (which failed), fleeing from the circus and vowing never to laugh again.
Archchancellor Ridcully has a similar lack as Granny, with the result that his jokes are one of many aspects of his personality that grate on the Bursar, who also has no sense of humour, but at least understands how jokes are supposed to work. In fact, when Ridcully first appears in Moving Pictures, the Bursar (who has not yet been driven completely insane by the man) attempts a joke himself:
The Archchancellor gave him a look so old-fashioned it might have belonged to an ammonite. "That a joke?" he said, in the suspicious tones of someone who wouldn't really understand the term "sense of humour" even if you sat down for an hour and explained it to him with diagrams.
Dwarves are known for their sense of humor. In a way. People point them out and say "Those little buggers haven't got a sense of humor."
Sergeant Detritus, a troll on the A-M City Watch, "treats humor as a human abberation" to be overcome by talking slowly and patiently.
Harry Potter: Percy Weasley "wouldn't recognize a joke if it danced naked in front of him wearing Dobby's tea cozy." He does have a bit of a harshly sarcastic moment in Deathly Hallows, though, which is remarked upon with astonishment by his siblings.
From Mercedes Lackey's SERRAted Edge novels, Unseleighe psychics have absolutely No Sense of Humor, along with little comprehension of allegory and the imaginations of bricks. In one novel, they were driven insane trying to comprehend the lyrics of They Might Be Giants.
From Animorphs, Andalites. All of them besides Ax and Arbron.
Thomas from Malevil. He even notes that he's known for not having a sense of humor but doesn't argue the point.
In Warrior Cats, Rainflower begins scolding Crookedjaw and Oakheart for telling friendly jokes about the queens, making a (quite) boring lecture about how the queens "enjoy helping their Clan". Crookedjaw and Oakheart just roll their eyes at this.
Colonel Lieutenant after promotion Brendig in David Edding's Belgariad. Also Ortzel from the Elenium. The two never smile. Ortzel does later, albeit it looking unnatural on him. Brendig does show some humour, but since this is a work of Edding's, compared to the others, his humour is close to, if not, nil.
Ce'Nedra:Don't you ever smile?
Brendig:(perfectly straight face) I am smiling.
In a series full of deadpan snarkers, Baron Vengeous from Skulduggery Pleasant has no sense of humour and an aversion to humour in general. Skulduggery's constant wisecracking is one of the things he hates about him. Interestingly though, in the second book Vengeous does crack a joke, saying "Forgive the expression but you and and what army" and at one point almost laughs at one of Skulduggery's jokes. Generally though, he's dour and humourless.
In 1066 and All That, Queen Victoria "remained obdurately plural and not amused" throughout her reign despite the best efforts of her subjects to amuse her.
A Song of Ice and Fire: Stannis Baratheon is perceived as this by the people of Westeros. The reality is that he has an extremely dry and subtle sense of humor which is lost on the average Westerosi who is accustomed to bawdy and/or toilet humor.
Vulcans are renowned for having No Sense of Humor, though many of them are Deadpan Snarkers instead.
Data from Star Trek: The Next Generation repeatedly attempts to understand humor as part of his quest to become more human. He doesn't succeed until he gets his emotion chip, which is one of the plot threads in the above-quoted movie.
Worf from the same series has almost no sense of humor either. He's always serious. Martok eventually works out that he has a very dry and sarcastic sense of humor, he just almost never laughs.
Lampshaded on Criminal Minds: Reid (falsely) calls Aaron Hotchner a narcissist as clue to where he is taken hostage. While figuring this out, Hotch asks the team what they consider his greatest folly. Spontanous answer by Prentiss: "You have no sense of humor!" Actually something of a subversion. Hotch does have a sense of humour; it's just so dry the Sahara looks like a swamp by comparison, and at this point Prentiss hasn't caught on to it yet.
"Now, nobody likes a good laugh more than I do... except perhaps my wife and some of her friends... oh yes and Captain Johnston. Come to think of it most people likes a good laugh more than I do."
Sheldon from The Big Bang Theory. He understands the concept of humor, he just doesn't get humor. When he's told that something is a joke, he forces the most fake laugh ever heard on television. And the things he does find humorous are usually beyond the scope of "mere mortals" like the remainder of the cast. Not until the end of the fourth season does he tell any jokes that get laughs (although that's partly due to Tough Room) and when he did so, he wasn't even aware he was doing it.
In early episodes of Mork and Mindy, Mork identifies people making In-Universe jokes by saying "Oh, humor — Ar-Ar!" And almost as often he does that when people aren't joking.
A common trait for the villains on Doctor Who. And some non-villains, like the Ood.
Major Neuheim of Private Schulz is a Pointy-Haired Boss, not to mention a Nazi, who is totally lacking in warmth and anything resembling a sense of humor. However, his humorlessness is itself a source of comedy.
Played with in Stargate SG-1, Colonel O'Neill has very specifically stated that his name is spelled with two Ls on several occasions; this is because there's another Colonel Jack O'Neil in the fictional version of the USAF, one with No Sense Of Humor. This is an in-joke and lampshading of the fact that Kurt Russell's version of the character in the movie (spelled with one L in the credits) was significantly less funny.
Teal'c almost never tells a joke or laughs (unless you count his occasional plays on earth idioms), with one exception. It turns out his sense of humor just doesn't translate cultures well.
Admiral Brovo: Don't worry, Teddy. Junior officers get thrown to the sharks first. And lieutenant Rabb is the one who's sailing into harm's way.
Commander Lindsey: Yes, sir.
Admiral Brovo: Harm's way? Lieutenant Rabb's first name is Harm.
Commander Lindsey: Oh! Oh, yes, sir.
Admiral Brovo: You need a sense of humor, Teddy, otherwise people are gonna think you're a lawyer.
Commander Lindsey: I am a lawyer, sir.
People who don't "get" Mystery Science Theater 3000 - a show that plays bad movies and heckles them as they play - sometimes are dealing with this. They find the riffing gets in the way of their enjoyment of the movie being played. Not everyone has this issue though - some people have a good sense of humor but it's hard for them to follow two things - the movie and the riffing.
In one of the Muppet's "at the ball" sketches, a female dancer asked her partner, "Do you know who is the most humorless man in the world?" When he said no, she replied, "You are, Fathead!" He, of course, responded, "I don't get it."
Batman in Batman: The Animated Series is thought by many to have no sense of humor, but that's a false perception influenced by his extremely introverted personality.
Adult Ben in the Ben 10 episode "Ben 10,000". He gets better though.
Optimus Prime in Transformers Prime. It's stated that all Primes are like this — the Matrix of Leadership, which contains the accumulated wisdom of all past Primes, gives them a sense of responsibility, dignity and gravitas that leaves little room for goofing around.
Optimus notably wasn't like this in the original series. While usually serious he was not above playing a game of basketball with his men, or cracking a joke or two, or mocking Megatron. However as the years and series went on his characterisation became more and more about being a no nonsence messiah type figure.
Lemongrab of Adventure Time. He tries, though. He really does try... but he fails horribly.
Lemongrab: "Ha. Ha ha ha! HA HA HA HA HA HA HA HA HA! Twelve years Dungeon! All of you! Dungeon! Seven years, no trial!
Zane in Ninjago is noted at trying to understand humor but seems to fail.The reason for this is that he is a Ridiculously Human Robot that normally has all of his emotion switches off, including humor.Its sadder then it sounds.
The Simpsons: Rex Banner never laughs. When he tries to do so, in "Homer vs. the Eighth Amendment", he fails miserably.
Germans are often stereotyped as having No Sense of Humor. In reality, German Humour is a bit more complicated; Germans definitely do have a sense of humour, and enjoy comedy, but their ideas about what is funny and what isn't can vary greatly from those of English speakers.
And German people think the English have no humor, or at least this special "English Humour"...
This trope was parodied by UN Ambassador Hans Beinholtz whose sense of humour was...dry, to say the least.
In Greece, if someone tells a joke that completely fails to amuse, people are likely to comment 'That's an English joke'.
An executive for Looney Tunes, Eddie Selzer, once overheard a group of writers laughing at jokes being thought up for a specific Looney Tunes. He stormed in and asked what all the laughing had to do with the making of animated cartoons.
He was well-known for having the sense of humor of an underripe potato. They used to use him as a sort of barometer for the quality of their ideas; if he liked it, they were off the mark. Selzer was, in fact, the specific genesis of the cartoon "Bully For Bugs", when he barged into a brainstorming session and declared, sans all provocation, that bullfights weren't funny, and there were to be no cartoons about them. They made the cartoon anyway, the argument being that Selzer had, in Chuck Jones' words, "never been right about anything yet."
Similarly, at MGM Tex Avery was forced to work for executive producer Fred Quimby, who, according to the book "Tex Avery: King of Cartoons", never understood any cartoon Avery made. Even though everyone else, including the audience, considered them to be hilarious. He even warned Avery not to make "Blitz Wolf", in which Hitler was ridiculed, "because we don't know who will win the war in the end."
The Marx Brothers: Groucho Marx always claimed that actress Margaret Dumont, who was always the Butt Monkey of his insults, never seemed to understand his jokes. Recent research has found out that this was simply an urban legend.
A frequent side effect of Asperger's is the inability to understand humour, as well as sarcasm, due to their being Literal-Minded. They might understand it but have trouble distinguishing it from sincerity, which leads to similar reactions. Not being so naturally keen on social interaction aspies can often develop a rather dark sense of humour, as they do not have the same inhibitions against it. This of course varies from individual to individual.
Which is rather odd considering that there are quite a few hadiths about Muhammed himself cracking jokes.
Attila the Hun is reputed to have had no sense of humour. Allegedly an envoy from the Eastern Roman empire who met with Attila once at dinner described it as follows: “By mixing up the languages of the Italians with those of the Huns and Goths, [an entertainer] fascinated everyone and made them break out into uncontrollable laughter, all that is except Attila. He remained impassive, without any change of expression, and neither by word or gesture did he seem to share in the merriment ….” The impression left was that Attila lacked a sense of humor and cared about nothing but power.
A common misconception of Ayn Rand is that she had no comprehension of humor, despite a talent for devastatingly on-target satire. She was sometimes Late to the Punchline, perhaps as a result of early trauma, and part of it could have been that she was a native Russian-speaker operating in an English-speaking environment; plays on words and things like that may have just sailed right past her. She was prone to quips however and sometimes demonstrated a biting sardonic wit. For instance, whilst being pestered by a fly on a talk show, she commented "Perhaps it's protesting."
Isaac Newton is widely reputed to have had no sense of humour.