In various media, con artists and other crooks who pride themselves on their wits tend to dislike using guns. Usually, they also dislike carrying them; sometimes, they don't want to be anywhere near them.
It might be simple fear because guns are dangerous weapons and the conman might very easily prove to be far less proficient in a gunfight than his supposed victim. It may be pragmatism, as guns are often taken as evidence of intent to kill and will usually get you a longer prison sentence if you get caught, as well as increasing the chance that someone will draw a gun on you in return. On the other hand, it is quite often a point of pride — guns are beneath them. After all, they aren't called the conartist for nothing — a truly good conman makes people do his bidding with cunning. Just threatening to shoot somebody is inelegant, if not outright cheating.
From an out-of-universe standpoint, it's usually because the easiest way to turn a Loveable Rogue, The Family for the Whole Family, or The Pirates Who Don't Do Anything into terrifying, evil criminals is to show them terrorising people with guns.
If guns have not been invented, then sometimes the con artist will have a dislike for swords, or knives, or whatever else is the most prolific/dangerous personal weapon.
Lupin III takes after his grandfather (in literature) by also being such a Consummate Professional that the gun is his last resort. The difference with his ancestor is that the Third still considers it a tool to be used. He also specifically recruited Jigen to his team because he is explicitly one of the — if not the — greatest gunslingers alive.
The English conman/sorcerer John Constantine from Hellblazer hates using guns, and will tend to use wits and magic even in the most direst of situation. This was later changed in the movie adaptation, where John is a gun-totting American exorcist.
Films — Live-Action
Roy in Matchstick Men keeps a gun next to his money, but when someone tries to rob him, he doesn't want to use it.
None of the cons in the Ocean's Eleven films like to use guns. The only time they do is when they're impersonating a SWAT team. In the third film, Linus shows professional disdain when a self-proclaimed master thief robs him with a gun. After taking his loot, the master thief reveals that the gun was empty.
In The Flim-Flam Man, the rather elderly conman Mordecai Jones (George C. Scott) never uses force and surrenders peacefully when a gun is pulled on him.
Layer Cake: the main character, a career criminal, expresses distaste for his comrade's gun collection, then immediately picks one up and starts fawning over it.
Captain Jack Sparrow of the Pirates of the Caribbean franchise will usually go out of his way to avoid killing or violence, preferring to sneak and con his way out of situations; as noted by Barbossa in the first film: "Now you see, Jack, that's exactly the attitude that lost you the Pearl. People are easier to search when they're dead." To the best of my recollection, he's only ever actually shot someone once in any of the four movies.
Done for multiple reasons in The Italian Job (2003). Charlie and company make it a point of pride to not use guns, but we also get the impression that they're not very violent people in general, and wouldn't do very well if it came to a shoot-out.
Mick Connelly in Betrayal in Death is a con man, pick pocket, and various other thievery-related professions. He expresses a disdain for the guns in Roarke's private weapon collection, instead perusing the knives.
Moist Von Lipwig in Going Postal dislikes swords because they "raise the stakes too high". In Making Money, he buys a cosh because life's too dull and he wants to raise the stakes a little, and immediately wonders what he was thinking. Vetinari also calls him on this, marveling what Lipwig "who has never struck a man" would want a weapon for. In Going Postal, Lipwig himself takes some pride in his non-violent nature and is offended when his golem parole officer calculates he has "killed 2.338 people" by ruining people and hastening the deaths of many by just a bit. At the end of Making Money, Vetinari experimentally shows that Lipwig dislikes weapons so much he is more nervous when holding a sword than when being threatened by one.
On another note, the Thieves' Guild prefers to incapacitate or frighten its victims into handing over their possessions rather than killing them outright, because that would intrude on another Guild's territory (the Assassins' Guild). Reducing the number of people there are to rob in a city is also bad for their business. Of course, they are free to punish those who steal without a license however they wish. Thieves breaking demarcation by killing people earn a short word from an Assassin: "Goodbye."
James "Slippery Jim" diGriz, the main character and con man from The Stainless Steel Rat series is a variant of this. He sometimes carries a .75 caliber recoilless handgun with explosive rounds; however he hates killing, only using the gun to Shoot Out the Lock and related things. He also carries it in the shower.
Arsène Lupin, in Maurice Leblanc's short stories. Very much due to the Consummate Professional aspect of the trope. He dislikes anyone risking their lives (despite putting his own on the line), whether the person taking the risk is one of his accomplices or his victim. Defying his rules can earn you a meeting with the hangman, as he withdraws his protection from the cops.
The villainous triumvirate in the Hand of Thrawn duology has one character who never makes threats or so much as touches a blaster, and that's the professional con man/actor. Then again, he usually gets on okay with the strategist, who is a clone of a Red Guard and is much more wiling to try violence. The politician once tries pulling a blaster on the strategist and is disarmed quickly enough that the strategist didn't even take the threat seriously; the con man probably remembered that the strategist is ridiculously good at combat.
In Time Scout, Skeeter Jackson doesn't hate guns so much as he'd prefer a weapon he's more familiar with, like a Mongolian recurve bow.
Jack from the Dragonback books, like his uncle, never carries any lethal weapon. After all, they can't accuse you of assault with a deadly weapon if you don't own one. He does, however, enthusiastically embrace nonlethal weapons such as tanglers.
In Lawrence Block's Bernie Rhodenbarr series, Bernie, a burglar and conman, fears guns, as he tends to picture them being used on him.
White Collar: Neal Caffrey is usually able to talk his way out of a situation and never has to use one. He does prove in one episode that not liking guns doesn't mean he can't be a very good shot with one, though.
Eliot doesn't like guns, but explains it's only because they're too imprecise (this from the guy who could probably kill you in various unpleasant ways with a toothpick or something). His reasons vary from episode to episode, but the real reason he doesn't like them is that they take him back to a part of him he doesn't like. However in The Big Bang Job he does state that not liking guns doesn't mean he can't use them. He is even able to put most action movie heroes to shame with his over the top shootout.
The rest of the team is a more straight example, besides the pilot, none of them wield guns for the entirety of the series. However in the pilot, both Parker and Hardison bring guns to the meeting when they had been cheated and in their flashback Sophie and Nate shot each other.
In one season finale, a really ticked off Nate practices using his father's gun, intending to shoot the man who killed his father. Everyone else on the team tries to persuade him not to. He doesn't end up shooting him, choosing to instead turn his partner against him and have them kill each other..
In The Mentalist, titular mentalist Patrick Jane is well versed in con games and general messing with people, and makes his dislike for guns apparent on several occasions - contrasting him with the team of hard-boiled state agents he assists. When two CBI teams have an unexpected meeting and reflexively draw their guns on one another, Jane yells at them to put them away, exclaiming "Cowboys!" in alarm and exasperation. He also immediately drops the shotgun he used to kill Dirty Cop Tanner, who was about to attack Lisbon.
In Warhammer Fantasy Roleplay, followers of Ranald, god of thieves and gamblers, also avoid violence when possible, both because it's crude and unprofessional and because murder is a premise of a competing god.
In Left 4 Dead 2, the conman Nick says that it's illegal for him to hold a firearm due to being a convicted felon.
Walter Hardy in The Spectacular Spider-Man typically doesn't carry a gun. The key word is typically — as he gets older and slower, he resorts to carrying one... and killed Ben Parker with it. He is genuinely remorseful, and believes he deserves to stay in prison, even turning down the chance to escape in "Opening Night".
Some criminals avoid guns because the prison sentence for possessing a gun during a crime is much greater than without one. Other times it's simply because physical confrontations aren't a part of their plans.
Some retired pickpockets have told reporters that any thug can threaten someone with a gun, but it takes skill to steal someone's wallet and not have them realize it's gone.