Lots of emotionless characters have been written and people who tend to be less emotional exist in real life and depending on their emotions, can engage in a wide range of personalities. Sometimes, they are among the most popular characters in the show.
The best example is Spock from Star Trek who was raised by an alien race that was capable of strong emotions but culturally believed that demonstrating those emotions was taboo and instead would favor logic above emotional responses. That didn't mean Spock was incapable of acting on Emotions or didn't understand their value... they just chose not to show it (and would always give a logical motive for a seemingly emotional response). Spock worked great because he was frequently paired with Dr. McCoy, a person who would always bring the emotional argument and leaving Kirk to have to navigate between the two options. Most episodes focused on a moral dilemma that found that the emotional response was just as terrible as the logical one. But the perfect solution was somewhere in the middle (and involved the captain ripping his shirt or hitting them with a double fist punch).
Spock was so popular that every spin off series had a "Spock" like character who would be an outside observer who provided an alien view on the human condition. The Next Generation had Data, an android who was actually incapable of emotional responses and desired them... but was a talented artist even without them and was capable of understanding and imitating emotional behavior. Voyager had Tuvok, a Vulcan who was a security officer and thus not the scientist like Spock and later 7 of 9, who had a more outsider view of humanity than Tuvok to comment on. Deep Space 9 is the oddball in that there are plenty of characters willing to be the "outside commenter of the human condition" but they also tend to have strong emotional motives that back their logical decisions. Dax, the science officer, is a hedonist and when off duty enjoys pleasurable experiences. Quark, the bar man on the station, is a buisness man and is motivated by a desire to boost his bottom line. Odo, the head of station security, is motivated by a strong desire for Order but is a non-solid lifeform compared to the rest of the cast. All have very logical backgrounds but also have various levels of emotional desire as well.
In the Homage series to Star Trek The Orville, the character of Isaac portrays this role. He is a member of the Kaylon species, a race of androids who's creators died out. They are all machine logical like Data, but unlike Data, Isaac has no desire to experience emotions. He is motivated purely to by a desire to understand biological lifeforms as a whole and will often ask questions out of a motivation to find the logic behind the "illogical behavior". As the series moves on, it becomes clear that the Kaylon lack empathy, something all of Star Treks logical characters had in spades. Without spoiling too much, this lack of empathy was not just the Kaylon's foible, but the fatal flaw of their creators. And not only are the Kaylon capable of learning empathy, but can learn it quite quickly.
Dialog tends to be overly clinical in nature, especially paired with a science mind. They will be precise in giving information (such as declaring that Earth has an orbital period of 365.25 days, rather than 365 days). They will also use words in their correct meaning, not their vulgar meaning (for example, when the have an idea as to what is going on, they will declare "I hypothesize" rather than "I have a theory" because as everyone knows, theories are proven, hypothesis are testable but unproven until testing. However, it's not uncommon for people to use "theory" as a synonym for a deduction based on observation). In the right hands this can actually be humorous to the audience. Imagine the scene.
Captian: in the meeting room with senior staffers We have to solve this, [Logical Character], do you have any theories as to what is going on?
Logical: I have no theories.
the entire meeting room looks at Logical in shock
Logical: I do, however, have several hypothesises, but I would need to test those. Only then can I have a theory as to the nature of the phenomena.
Captain: exasperated Fine, whatever. Care to share with the rest of us?
Humor can also come in other ways. Sometimes the "logical" thing to do is to have an emotional discussion. Suppose a scene where Logical Character comes upon an injured person and has knowledge of first aid treatment:
Logical: While treating the victim Tell me about your romantic partner or what you perceive are desirable features of a romantic partner.
Victim: Gives Logical an answer and asks Logical for his preferences
Logical: I have no desires for a romantic relationship. I only inquired so that you would discuss a topic other than your injury as such conversations will help keep your mind off injuries which can prevent you from going into a state of shock. In our present situation, such a reaction would only complicate the matters further. There was no desire to learn of matters of a personal nature to yourself. Beat She sounds like a fascinating person.
Emotionless characters will tend to not have the ability to make interesting small talk and it can be difficult for people who actually want to get to know them to breach subjects with them when they have a difficult subject to discuss and want to ease into it. The emotionless character might not be candid with small talk, but once the point is reached, they will give a direct answer and explain the reasoning behind it. They may also respond to conversation starters in this manner. Asking them about the weather will result in them repeating the most recent weather report. Asking them "How's your day been" can have a neutral response if the day is not unusual for the character's routine.
Another thing to look for is the "Mathematician's answer" which is an answer that satisfies the question, but doesn't actually provide information that the person asking the question is seeking or finds useful. Consider the scene from Marvel's Avengers where Iron Man and Captian America are trying to fix the engine and the former has to talk the later through it. Iron Man asks Cap to open a panel and describe what he sees. Cap does so, revealing a complex mess of wires and lights... he replies "It appears to run on some form of electricity." Iron Man quips that "Well, you're not wrong..." Obviously, he wanted to know something more specific so he could tell Cap how to fix it... but Cap's so out of his depth he can't even give a description that could hope to help Tony... but what he does give him satisfies the answer and is correct.
The other example can be found in the misuse of the logical AND and OR in place of the linguistic "and" or "or". In logic, any statement that joins two or more logical statements by AND are true if both statements are true ("There are no clouds in the sky AND the sky is blue" is false if there are clouds in the sky. It's also false if there are no clouds in the sky BUT it's night time, so the sky is black). By contrast, any statement that joins two or more logical statements by OR is true if any of the logical statement is true. ("There are no clouds in the sky OR the sky is blue" is true if it is partially cloudy during the day because the sky is still blue. It's also true at night if there are no clouds in the sky, but the sky is black).
Someone can ask the Logical character if he likes blondes or brunettes. To which the logical character will respond "Yes", which is valid if he prefers either blondes only, brunettes only, both equally or if it does not affect his preference at all. His answer of "Yes" tells you nothing, but satisfies the question being asked to him.
This second form is often the logical character does have a sense of humor or has some sense of humor, because the person giving a such an answer has to have a good understanding of language and how words have multiple meaning, understand the desired answer of the question, and respond in a way that frustrates the person asking the question (often because they're frustrated by the question being asked in the first place as it's not required at the moment. Therefor the response should be just as useful as the question).
It should be pointed out that someone without emotion is not without passion. Spock and Data both had personal goals and interests they were trying to achieve. In fact, a neutrally toned, "Fascinating" was practically Spock's catch-phrase indicating that he was interested in something or had a eureka moment. Not to mention that he does have strong desires to procreate. Data, in his free time, would engage in theater, arts, and music. Isaac didn't want to just understand the unusual behaviors of biologicals, but wanted to participate and experience them (while he was baffled by the concept of a practical joke at first, once another character explained the practice, Isaac not only participated in the prank war, but was declared the winner of it for a prank that involved amputating someone's leg without their knowledge (scifi medicine made this less horrific than it sounds, as the ships doctor could regrow amputated limbs)). However, all of them were logical outgrowths of their character... not based on emotional whims. Passion might not be the best word, but you could feel the enthusiasm each character received from participating in the thing. Going back to Data, in one episode he is given several gifts, and proceeds to unwrap them in a delicate manner so that the giftwrapping may be used in the future. When another character points out that "that's not the point" Data rips the paper up and tosses it to the floor... despite the gift already being removed from the wrapping... he still doesn't get the point, and knows he missed something by the laughter of the person who protested... but understands it made the friend happy that he at least tried to meet expectations even without understanding it.