You might be interested to learn about the Italian commedia dell'arte, a form of theater that was popular from the 16th to 18th centuries. It was a form of improv/sketch comedy based on stock characters and familiar situations, and has been extremely influential on Western comic culture.

There were three types of characters. Servants (called zanni, which our word "zany" comes from), Masters (old, stupid men), and lovers (young men and women). Many of these characters were stereotypical caricatures of various Italian cities--the pretentious doctor from Bologna, the stingy merchant from Venice, the boastful yet cowardly soldier from Naples or Spain. The plots were simple and recognizable, like trying to get out of a debt, or seduce someone's wife, but the details would be improvised on the spot, featuring clever wordplay, banter, and slapstick (a slap stick was a device called a batacchio, a fake club with two wooden boards that would slap together, making a very loud sound but not actually hurting). Most commedia dell'arte was performed by troupes that would travel around, and actors would usually specialize in one character for their entire career, sometimes only appearing in public in costume (like Sacha Baron Cohen).

Here's a clip of the famous comic Dario Fo doing an Arlecchino routine: https://www.youtube.com/watch?v=jvC-y4I_5hU (start at 00:20 to skip the introduction). Arlecchino was a servant character whose schtick was being always drunk and/or hungry. He usually wore a patchy, multicolored outfit, and is the origin of the term "Harlequin."

Radio shows are a predecessor to be the first TV sitcoms, however recurring cast/characters in situational comedy were used by playwrights since ancient greece; Shakespeare wrote several. So to answer your question there were similar forms long before Farnsworth was even born.

The responses here are very good, but there is one more thing I would like to add: comic strips feature a very similar format to TV sitcoms, and could also be included in the list.

Also, this is not directly related to sitcoms, but in the pre-TV days it was very common for novels to be published in series, with one chapter being released at a time, in the same way modern television shows work. Most of Charles Dickens's works were published this way.

"Punch and Judy" shows were a common English form of puppet theater that you'd see at fairs, festivals, and sometimes just in busy streets. Punch, a short-tempered hunchback who frequently whacks other puppets with his stick, his wife Judy, who berates Punch for his stupidity, ineptitude, greed, or gluttony, and a host of other regular characters, all known to the audience, may appear throughout the show as well. Punch even has a distinctive voice, normally produced by a device called a swazzle held in the performer's (known as the "Punchman") mouth.

You can still see Punch and Judy shows at Renaissance fairs and other festivals today.

You might be familiar with 'film serials' of the 1940s, usually they were action-adventure ones, but series of short films like Joe McDoakes, Three Stooges or Our Gang/Little Rascals could be considered proto-sitcoms.

In 16th century Italy Commedia dell-arte was a live theater performance style with the same archetypes playing out different scenarios. There were wise servants, the young lovers, the miserly rich man, the absent minded scholar and a proud but cowardly military man. I will add more and cite sources when I get home to my books.

The average vaudeville act in America was ten to fifteen minutes long, many of the early early TV pioneers got their start in vaudeville. This type of entertainment faded due to the invention of radio and sound film, but the dance routines, short musical numbers, jokes and gags of vaudeville lived on and adapted them to the new formats.