In general, the show was good about keeping the years straight, but it occasionally broke down in this regard.
In one episode, both Gojira and The Blob are referenced. Neither of those movies were released during the Korean War (Gojira: 1954/Godzilla, King of the Monsters: 1955, The Blob: 1958).
In "Der Tag," Radar is shown sleeping with a copy of The Avengers on his chest, with the 1970s' logo. One shot later, it switches to another issue of the same comic with the '60s logo.
In "The Novacaine Mutiny", Radar also has a Spider-Man comic.
The Points system is referenced several times, and is how Blake and Painless Pole are discharged. The Points system was discontinued for rotation of personnel several years before the Korean War, and was never used for surgeons.
BJ's latter-seasons hairstyle was much longer than a professional man in the 1950s would have worn. (This may be excusable, since the whole point was that BJ was rebelling against the Army.)
Several times, Korean soldiers are shown with AK-47-type rifles (actually stand-ins) before any communist nation even issued them yet.
In one episode Klinger hands out Hershey bars with UPC symbols on the back wrapper to recovering patients.
A pinball machine from the 1970s appears in the Officers' Club.
Henry uses a bullhorn that wasn't invented until after the war.
In "Officer of the Day," Flagg appears wearing the branch insignia for military intelligence. This insignia wasn't used by the Army until 1962.
Medics are always shown wearing helmets with the Red Cross painted on the front. This practise was stopped in early 1951 because North Korean snipers were using them to aim.
General MacArthur is almost constantly referred to as the Allied Commander. MacArthur was relieved of command in Spring 1951.
Colonel Blake had a framed photo of a UH-1 Huey helicopter. The Huey first flew in 1956.
Hawkeye and BJ are dying to see The Moon Is Blue, which was released on July 8, 1953, just three weeks from the end of the Korean War.
Klinger mentions he would wear Hula Hoops in his ears if he could. Wham-O first introduced it in 1958, and only started calling it that in 1962.
Radar's impersonation of John Wayne ("I'm not gonna hit you. Like hell I'm not!") is from McLintock!, released in 1963.
Grenades, more often than not, are painted flat black, as a subtle reference to Vietnam. In Korea, grenades were always olive drab.
In "The Gun", the revolver in question is a Colt New Service, in .45 ACP. The New Service was first introduced in 1898, yet either the frame or bone grips have an inscription with the date 1884.
The pilot of the chopper that flies Hawkeye out of the compund wears Nomex flight gloves. These were not used until 1967 during the Vietnam war. In Korea, pilots wore black leather gloves.
Virtually all clothing are cuffless permapress fatigues. Neither cuffless nor permapress utilities were available until the 1970s.
In "Bombshells", everyone is ecstatic about Marilyn Monroe is coming to the camp. While she did visit troops in Korea, Marilyn Monroe's big leap to stardom did not come around until Niagara in January 1953, while the camp had to make do with older films and film that new would not have reached them that late in the war. Before then, she was still relatively unknown.
In the same episode, Charles mentions "There's no business like showbyness" when pointing out a misspelling of Marilyn on a banner. While it could be a reference to Annie Get Your Gun, where the phrase "No business like show business" originated, it could just as easily be (and more likely is) a reference to the 1954 Marilyn Monroe film of the same name.
One examples teeters precariously on the borderline between averting this trope and being another egregious example. In one scene, Charles brags to Hawkeye and BJ that he once had dinner with Audrey Hepburn. Hawkeye and BJ scoff at the idea that such a famous beauty would have anything to do with Charles...until he shows off a photograph of the two of them together. The problem is that Hepburn didn't become a truly major star until the release of Roman Holiday, a film that came out exactly one month after the Korean War ended. However, she did have some measure of fame during the War. She starred in the original Broadway production of Gigi from 1951-52. So it's at least possible that Hawkeye and BJ had heard of her during the War, but given where they're from (Hawkeye's from Crabapple Cove, Maine, while BJ's from Mill Valley, California) it's extremely unlikely.
Within that same scene, Charles states he's never seen any of her movies. She had been in some movies by that time, but mostly in bit roles...with one big exception, where she had a major supporting role in the 1952 British film, The Secret People. Once again, possible but unlikely someone during the war would refer to her movies.
In "Soldier of the Month", Hawkeye and BJ walk past a sleeping soldier by the mess tent. Hawkeye points to him and says "Used to be Albert Anastasia's doorman," joking that the man was on guard when Anastasia was murdered. Albert Anastasia was murdered in 1957, four years after the Korean War ended.
In "Soldier of the Month", Potter mentions the soldier of the month was a directive by the War Department. After World War II, the Departments of War and of the Navy were merged into the Department of Defense.
Later on in the series, beer cans are '60s and '70s era pull tabs (patented 1956 and first available in 1964 on Royal Crown Cola drinks) and modern aluminum stay tab cans not available until around 1980. Earlier seasons had the correct three piece cans opened with churchkeys.
In "Deluge", "Phisohex" bottles are seen in the "scrub" area. Phisohex was a cleanser used for antibacterial cleansing as well as an anti acne regimen. However, they are in polyethelyne containers not available until the 1970s. In the same episode, BJ is shown removing sterile latex surgical gloves, which were not developed until the early '60s.
In "The Incubator", an unscrupulous colonel selling materiel tells Hawkeye and Trapper that he could procure a B-52 if given a week's notice. The B-52 Stratofortress first flew in 1952, but did not enter service until 1955. In Korea, B-26, B-29, B-47 and B-50 bombers were used in bomber duty.
In "The MASH Olympics", Potter uses a Smith and Wesson Model 19 snub to start the crutch race, which turns into the stainless version, the Model 66, when he starts the partner race. The Model 19 was introduced in 1956, while the Model 66 was introduced in 1972.
This is a bit abstract, but the political landscape is not quite accurate to the time period. American politics is portrayed as being divided between liberal pacifists (the good guys) and conservative warmongers (the bad guys), which is accurate to the Vietnam era when the show was made. However, almost all domestic opposition to the Korean War came from right-wing isolationists and actual communists (and yes, people at the time commented on the "strange bedfellows" situation). Mainstream American liberals supported the Korean War because (unlike Vietnam) it was being fought under the auspices of the newly-founded United Nations and was seen to promise a new era of left-wing internationalism. Essentially, Frank Burns is an anachronistic neoconservative and Hawkeye is an anachronistic New Leftist.
In "The Colonel's Horse", Radar spells out BJ's father-in-law's name, H-A-Y-D-E-N, to which Hawkeye quips M-O-U-S-E, parodying the Mickey Mouse Club song. The Mickey Mouse Club first aired in 1955.
In "Baby It's Cold Outside", while watching the Sonja Henie movie "Sun Valley Serenade" Colonel Potter says at one point of the movie "This is supposed to be where she does a triple axel and ends up in a split." Then he has to leave and says "Now I'll never get to see it." He wouldn't be able to see it until 1989 when Midori Ito from Japan was the first woman to complete a triple axel in competition, Tonya Harding was the 2nd woman to complete one and the only American woman. It's also unlikely Sonja Henie ever did something as complicated as a triple axel, even with her three Olympic gold medals.
In "Deluge", we are told that the Chinese have become involved in the conflict and have attacked with some 300,000 soldiers. This apparently refers to massive attacks by the Chinese starting on November 1, 1950. Although the numbers are correct, the US/UN forces were unaware of the numbers. As late as November 6, the Far East Command continued to insist that there were no more than 34,500 Chinese communist soldiers in country.
In "Souvenirs", the MP describes the antique vase Burns is supposed to have bought as "An 800-year-old seladon vase of the Ko-Yu dynasty". Later, Burns packs up a white vase to send to his wife. Seladon is by definition green, and that kind of color glazing definitely wasn't around in the 13th century. What Burns packs up looks most like early to middle Quing period - or rather a contemporary ripoff.
In "Last Laugh", when General Fox points to the star on his shoulder and asks Hawkeye, "What does this mean to you?" Hawkeye responds by quoting a famous Texaco advertisement rhyme, "Always trust your car to the man who wears a star." However, Texaco did not introduce this advertisement jingle until 1962, 9 years after the Korean War ended.
Major Houlihan tries to contact her husband at the "2nd Army" in Tokyo. The Second United States Army was a World War I and World War II formation and was not in Korea.