The earliest example of the "tied to the train tracks" trope presented on stage was "Under the Gaslight", an 1867 play by Augustin Daly. Interestingly, it was the hero who was tied to the tracks, with the heroine saving him at the last second before the train zipped by. The most popular forms of theater at this time were melodramas featuring some kind of grand spectacle or sensation as its climax, from buildings on fire, to tying the hero to a piece of lumber as it is fed into a sawmill. Trains were a common element in many of these spectacles; Charles Bolton's "The Engineer" in 1863 featured the villain being crushed by a locomotive that narrowly misses the heroine.
"Under the Gaslight" was followed by Dion Boucicault's "After Dark" in 1868 with a railway rescue scene so similar that Daly sued the production for plagiarism. In the resulting lawsuit, Boucicault alleged that Daly had stolen the idea from "Captain Tom's Fright", a magazine story published a few months earlier in 1867. In that story, the hero (again, not a heroine) is knocked out by his mutinous crew and tied to the train tracks. He is not rescued though; rather, we discover at the last second that the train passes by on a neighboring track.
This lawsuit was an interesting development in copyright law, as until then, infringement was only considered in copying complete works. This lawsuit was concerned with infringement of a single scene within a larger work. Because the court record contained the contemporaneous research undertaken by both parties in this lawsuit, we can be reasonbly certain that Daly's play did, in fact, contain the first staged scene of a character tied to the railroad tracks and rescued at the last second.
The success of Daly's play in the US and Boucicault's in England meant that this infamous railroad scene would be copied on both sides of the Atlantic on many stages. By 1868, no less than 5 theaters in London featured a play with this spectacle (some of which actually involved a woman tied to the tracks rather than a man).
The 1870s saw the birth of the Realism movement in theater, which in many ways was the antithesis of melodrama and spectacle. It grew in popularity to the point where it had displaced melodrama. I think it's important to point out that as we head into the twentieth century, melodrama is viewed as "lesser than" realism; they are seen as the dusty, silly shows your grandmother watched, as opposed to the sophisticated shows of Ibsen and Chekhov. It would not be until the 1960s and 1970s that serious academic study of stage melodrama would take place.
The silent film era ran roughly from 1894 to 1924, with the height between 1910 (the birth of Hollywood) to 1920. Despite the image of a mustachioed villain in a top hat tieing a damsel in distress to the train tracks, only to be rescued at the last second by our hero, this sequence does not ever really show up in serious silent films. The earliest existing example is Edwin S. Porter's "Train Wreckers" from 1905. However, it lacks many of the components of the trope. In the film, a woman is knocked out in revenge by a group of criminals and left unconcious on the train tracks. She is not tied up, however, and she is lifted off the tracks by the train engineer as it reaches her.
"The Open Switch" (1913), is another example that does not quite have all the elements of this trope. The heroine, Grace, is tied up near (not on) a set of train tracks as the villain plans to crash a train in another location. However, Grace manages to free herself and save the day.
Some point to the 1914 "The Perils of Pauline", starring Pearl White, as an example of this trope, but it never actually happens in any of the chapters. She is tied up by cowboys in a cave at one point, but never to the train tracks. However, a 1947 film also called "The Perils of Pauline," does have a brief flash of this trope. This film is a fictionalized biopic of Pearl White, and is a broad satire of the silent-film era rather than an accurate historical recreation. Much like how fans of theatrical realism looked down at melodrama stage plays, so too did post-World War II film audiences look down on silent films. They were simplistic films, filled with tired conventions and a lack of subtlety.
This trope does appear in some comedies. However, these were broad satires of the Victorian melodramas, much like "The Perils of Pauline" (1947) was a satire of the silent-film era. Mack Sennett, the producer responsible for the Keystone Cops and other well-known slapstick films of the era, produced some of the earliest examples of films using this trope. The first is the 1913 silent film "Barney Oldfield's Race for a Life". This one has it all: a villain with an obviously-fake moustache and tophat, a damsel in distress tied to the tracks and rescued at the very last second. Another Sennett film, "Teddy at the Throttle" (1917) also features a woman, played by Gloria Swanson, tied to the train tracks. However, she rescues herself.
Ben Singer, who has exhaustively studied American silent serial films from 1913-1918, writes multiple times about the trope of "woman tied to the railroad tracks" and other such spectacles as being products of the Victorian stage melodramas of the late 19th century, rather than of film. The melodramas and action serials of this era rarely rely on these tropes other than for parody.
So despite this trope appearing very rarely in silent films (and mostly in comedies), it nevertheless became the go-to example by later audiences and critics of how unsophisticated and corny silent films were. From the 1947 "Perils of Pauline," to the broad animated satire of "Dudley Do-Right" in the late 1960s (the gag is played in the opening credits to every episode, as well as being a plot point in the second episode of season 2 of "Dudley Do-Right of the Mounties"). The image was so immensely popular during the heyday of Victorian stage melodramas, that it has stuck in our collective memory and become an instantly recognizable shorthand for pointing out how some older works of art had simple morality lessons, broadly drawn characters, and rigid gender roles.
A few sources:
"Daly V. Palmer, or The Melodramatic Origins of the Ordinary Observer", Bruce E. Boyden. Syracuse Law Review, Vol. 68, No. 1, 2018. http://lawreview.syr.edu/wp-content/uploads/2018/05/O-Boyden-Article-FINAL.pdf
"Sensation drama, the railway, and modernity," Literature, Technology, and Modernity, 1860-2000, Nicholas Daly, Cambridge University Press, 2004.
"The Nervous Stage: Nineteenth-century Neuroscience and the Birth of Modern Theatre", Matthew Wilson Smith, Oxford University Press, 2017, pp 87-93.
"Melodrama and Modernity: Early Sensational Cinema and Its Contexts", Ben Singer. New York: Columbia University Press, 2001.
"Silent Movie Myth: Tied to the Railroad Tracks", by Fritzi Kramer, March 12, 2013. http://moviessilently.com/2013/03/12/silent-movie-myth-4-tied-to-the-railroad-tracks/
The silent films mentioned are all in the Public Domain and can be viewed on YouTube and other sites. Likewise, the full text for the Victorian melodramas mentioned can also be found on numerous websites.