With contributions from Robert, Seve, EC, Eddie and BZ.
An anachronism is an object or a person or a thing that doesn't belong in the time period it is found in, like a Roman soldier using a machine gun. Any film or television series set in the past has to be careful to present the time period as accurately as possible. However, it is inevitable that some historical inaccuracies will take sneak in. For example, a film set in the early 1960s using music from the late 1960s would be considered anachronistic.
The producers of M*A*S*H worked hard to make the series feel like it was set in the early 1950s, but it is understandable that a few anachronisms would slip in. Usually, these mistakes would take the form of a passing reference to a film or some other aspect of pop culture not realized until after the Korean War ended in July of 1953. Pointing out these anachronisms is not meant as a slight against the series.
Browse through the following list of anachronisms discovered so far:
In "Mail Call Three" Radar states that his mother's boyfriend takes her to bingo night in his Nash Metropolitan, not introduced in the United States until March of 1954.
At the end of "A Smattering of Intelligence" Captain Stone uses a Minaux camera, which was not put into use until the 1960s.

The pinball machine in the Officer's Club is a Gottlieb "Spot-a-Card," not introduced until 1960, some seven years after the war ended.
In the Season Eleven episode "Give and Take," Klinger hands out Hershey chocolate bars to soldiers recovering in Post-Op. Universal Product Codes (or UPCs) are clearly visible on the back of the wrappers. UPCs were not used until 1974. In an earlier episode, Season Four's "The Bus," Frank pulls several chocolate bars from his knapsack and hands them out; no UPCs are visible.
In several episodes, various characters drink from aluminum beer cans, which weren't used until the late 1950s.
At the start of Season Four's "Der Tag," Radar is seen reading a copy of the Avengers comic book, which was first published by Marvel in 1963. Additionally, the issue itself switches from #72 (originally published January 1970) to issue #60 (originally published January 1969). Another mistake!
In the first season episode "Tuttle," Radar is seen reading an issue of Captain Savage and His Battlefield Raiders! which was originally published by Marvel beginning in 1968. The issue in question is #10, from January of 1969.

The Godzilla movies were mentioned, but the original Godzilla film wasn't released until November of 1954, more than a year after the war had concluded.
Another film, The Blob, was also mentioned, but it too wasn't released until after the war had ended; it was first seen in September of 1958.
Also, in "Movie Tonight," Radar does an impersonation of John Wayne, from the film McLintock!, which wasn't released in the United States until November of 1963.

In many episodes, a red banner/poster with the words "4077th MED CO AIR AMBULANCE" and the image of a helicopter with a red cross on it can be seen in the Officer's Club. However, the helicopter is not the famous type seen in the opening credits that was actually used in the Korean War. It may be a "Huey" chopper that was used in Vietnam.
In several early Season Two episodes, a model of a helicopter can be seen hanging from the ceiling in Henry's office. It appears to be the same "Huey" model on the banner in the Officer's Club (see above). According to posts by Larry Gelbart at the alt.tv.mash newsgroup on Usenet (see here, via Google Groups) the model was made by his son and removed once it was found to be anachronistic.