The public won't know that the removed one either screwed up or incurred the wrath of those more powerful. In either case, the removed director will live to direct another day. More rarely, a director will take the initiative and leave a production because of conflicts that cannot be resolved (usually attributed to "creative differences"), again usually with the studio or the star. All of this will most likely take place either before or during the first few days of shooting.
But wait, there's more! You may at some point have noticed the name @George @Spelvin or @Georgina Spelvin or G. Spelvin or the like among the acting credits on a film. That's traditionally the alias used by actors who for one reason or another do not want to be credited with their own names. (The original Georgina Spelvin was the star of a famous porn film). There can be lots of reasons for using the alias, ranging from unhappiness with the way the production turned out to conflicts with the director or producer, or simply as an in-joke. In the same way, directors have sometimes used "@Alan @Smithee" as their alias when they didn't want to be credited under their real name. In 1997 screenwriter @Joe @Eszterhas (best known for writing "Basic Instinct") even had the bright/stupid idea of making a movie called "An Alan Smithee Film." Not only did it bomb at the box office, but the real director -- @Arthur @Hiller -- took his name off the credits, so it truly was an "Alan Smithee" film.
Now to the below-the-line credits. When the movie fades to black, a new set of names begins to scroll up the screen. Sometimes the first credits we see will go to the production crew, the people who worked on the shooting, and sometimes they will be for the cast, often in order of prominence in the film, though sometimes in order of appearance or in alphabetical order. The production crew credits will be where all the "funny" titles start coming up, and here's what they do: The "gaffer" is the chief electrician.