The Cameo: A ghostly sartorial version. The studio wardrobe department had trouble getting just the right look for Frank Morgan's "Professor Marvel". Finally, they went to a nearby thrift shop and bought an old, shabby frock coat for him to wear. While on the set, Morgan turned out the pocket of the coat, and noticed a name tag of the previous owner: L. Frank Baum. (This was later confirmed by Baum's widow and the tailor that made the coat.)
Dawson Casting: Here it's a 16 year old Judy Garland playing an 11 year old Dorothy Gale. This is largely unnoticed unless the viewer has read the book. Shirley Temple was originally considered for the role, but the plan fell through.
Nowhere in either the book or the movie is Dorothy's exact age mentioned, though the book describes her as a "little girl." Some sources suggest Dorothy was meant to be fourteen in the film while others assume (from the casting of Fairuza Balk in Return to Oz) that in the book she is around eight.
Dyeing for Your Art: Margaret Hamilton could not eat with the make-up on (Bert Lahr also decided to have his redone daily as he got tired of a liquid diet). The Tin Woodsman costume worn by Jack Haley was so stiff that he had to lean against a board to rest. The Cowardly Lion's costume was really hot due to being made of real lion skin. And everyone had to suffer with really hot studio lights.
Mean Character, Nice Actor: Margaret Hamilton as the Witch provides a rather infamous example. She was a Kindergarten teacher, and children would ask her frequently after the film why she was so mean to Dorothy, to the point Hamilton guest starred on an episode of Mister Rogers' Neighborhood to explain that the witch was only a role she played. It's also reported that Judy Garland found it difficult to actually fear her.
Hamilton was also devoted to animal rights. Her presence in SPCA television spots was as ubiquitous as Sarah MacLachlan's today.
The Other Marty: Jack Haley became the Tin Man after the original actor (none other than future Jed Clampett, Buddy Ebsen) was hospitalized. The Tin Man's makeup originally consisted of aluminum powder, which coated the actor's lungs and nearly suffocated him. To avoid the same near-fatal mistake, the makeup was changed to aluminum paste.
It goes further than that, originally Buddy Ebsen was supposed to play Scarecrow and Ray Bolger was supposed to play Tin Man. Bolger, however, longed to play the Scarecrow, as his childhood idol, Fred Stone, who had inspired him to do vaudeville in the first place, had performed the role on stage in 1902. Because of this, he was unhappy with is role as the Tin Man, reportedly claiming "I'm not a tin performer; I'm fluid", and convinced producer Mervyn LeRoy to recast him in the part he so desired. Ebsen agreed to switch roles with Bolger.
In addition, the Wicked Witch of the West was originally supposed to have been played by Gale Sondergaard and the character was originally supposed to be a glamorous witch inspired by the wicked Queen in Snow White and the Seven Dwarfs. However, when producer Mervyn LeRoy decided that having an attractive Wicked Witch created a plot hole, as it played against the notion that (as "Glinda, the Witch of the North" would eventually point out to Dorothy) only bad witches were ugly, the character was made into the familiar "ugly hag" and Sondergaard, looking hideous in the make-up, left the production and was replaced by Margaret Hamilton.
Talking to Himself: Frank Morgan is most notable for having played Professor Marvel, the Gatekeeper, the Mayor/Carriage Driver, the Guard, the disembodied Oz head, and the Wizard himself.
Also, there are a few actors who have played numerous characters: Ray Bolger as Hunk and the Scarecrow, Bert Lahr as Zeke and the Cowardly Lion, Jack Haley as Hickory and the Tin Man, and Margaret Hamilton as Miss Gulch and the Wicked Witch of the West.
And there are other little people who have done both roles: both Fern Formica and Margaret Pellegrini as both Munchkin Villagers and the two sleepyheads, and Karl "Karchy" Kosiczky (now Karl Slover) as one Munchkin Herald and a third sleepyhead.
There are a few people who have voiced other characters, including:
Billy Bletcher as the Mayor of Oz and the Lollipop Guild member.
Lorraine Bridges as one Ozmite and the Lullabye League member.
And Abe Dinovitch as one apple tree and one of the Munchkins.
Troubled Production: Cast and director changes, budget overruns, impractical costumes and make-up (specially under heavy lighting!)... basically everything that could complicate the shoot happened.
What Could Have Been: The script originally included an end scene that was never filmed, in which Hunk (the real-world counterpart to the Scarecrow) went away to agricultural college and Dorothy promised to write to him. The implications were heavy that this would result in a romance between them, which would account for Dorothy's particular affection for the Scarecrow during her time in Oz.
Also, in addition to the Dawson Casting example, WC Fields was originally asked to play the Wizard, but he demanded a salary which MGM considered to be too exorbitant.
"The Jitterbug" scene and number, even though the finished film still has a line leading into it from the Wicked Witch ("I've sent a little insect ahead to take the fight out of them!")
"Over The Rainbow"note This title specifically. The "Somewhere" part wasn't originally a part of the title. was very nearly cut from the film because the producers thought it was disrespectful to have Judy Garland sing in a barnyard (and because it was thought that it would slow the movie down). Cutting that song would have changed her entire career.
Also there was originally meant to be a Dark Reprise of the song when Dorothy is trapped in the Witch's castle. As Judy Garland would have had to incorporate a lot of acting into the song, it had to be recorded live during the take. Reportedly it reduced the entire crew to tears. Here's the audio. Unlike in the above scene, however, the song was cut at this point.