Why do movie teams shoot alternate endings?

There is a wide variety of reasons as to why a movie might have alternate endings.

1. The original ending tested poorly during screen tests: In this case the director had a specific ending that made it through production, but was received negatively by test audiences, forcing the studio to re-shoot the ending to better please the target audiences. An example of this is DodgeBall: A True Underdog Story (2004) which originally had Average Joes losing at the end. (Discussed here)

2. The script wasn't completed when the endings were shot: in this case the details of the script are still being worked on, so the crew shoots multiple endings with the hope that one will eventually be used as the final, official ending.

3. The original ending was seen as too graphic or violent especially studio is trying to hit a specific rating (usually PG-13 instead of R), so the ending will be slightly edited: An example of this was Thelma and Louise (1991) (discussed here).

   Only a tiny tweak here, but a fairly significant one — the first ending showed Thelma and Louise's car tumbling all the way to the canyon floor, no doubt getting pulverized in the process. Harvey Keitel's character finds the Polaroid that blew out of the car and looks at it as a helicopter heads down into the Grand Canyon to survey the wreckage. As you probably know, the updated ending is a wee bit more hopeful — we see their car drive off the cliff, but not the aftermath.

4. The studio or the director specifically wants multiple endings: the only example I can think of for this is Clue (1985) where the movie was shot with three different endings, each one would be randomly selected during each showing. Leaving the audience guessing which ending they would actually get.

There can be many reasons for shooting alternate endings; a few of them are:

• To pass censorship:

  In 1975's Bollywood film Sholay the director had a different vision for the ending from the one we see in theaters but the Board of Censors didn't allow them to show the brutal killing of the antagonist, and made them re-shoot the scene as a police arrest of the antagonist. But they later showed the original ending on the UK satellite channel 'Zee TV'.

  From Wikipedia :

  The director's original cut of Sholay has a different ending in which Thakur kills Gabbar, along with some additional violent scenes. Gabbar's death scene, and the scene in which the imam's son is killed, were cut from the film by India's Censor Board, as was the scene in which Thakur's family is massacred. The Censor Board was concerned about the violence, and that viewers may be influenced to violate the law by punishing people severely. Although Sippy fought to keep the scenes, eventually he had to re-shoot the ending of the film, and as directed by the Censor Board, have the police arrive just before Thakur can kill Gabbar. The censored theatrical version was the only one seen by audiences for fifteen years. The original, unedited cut of the film finally came out in a British release on VHS in 1990. Since then, Eros International has released two versions on DVD. The director's cut of the film preserves the original full frame and is 204 minutes in length; the censored widescreen version is 198 minutes long.

  _{Related question: How many different endings does Sholay have?}
  
  

• To test the screening response:

  Sometimes directors shoot new endings to satisfy the negative responses during test screenings. The Butterfly Effect suffered the same fate too, where it was felt that the original ending was too dark, so they shot multiple new endings and selected one of them and made the other three available on DVD.

  _{Related question: What was the original ending and why was it changed?}
  
  

• As an extra feature for DVD:

  Some endings never made to the final scripts, or were cut and used in the end as added features for DVDs for sales purpose. Sometimes these alternate endings are not even shot and are presented as director/cast/crew commentary.
  
  

• Director vs producer/studio differences

  Salt had two endings because producers didn't allow the director's vision. From Wikipedia:

  The Director's Cut was described by Noyce as "my own personal take on the material, free from the politics and restrictions of producers, studio or censorship ratings."

  _{Related question: Why does Salt have two different endings?}

To avoid leaks - Game of Thrones springs to mind

During a recent talk at his alma mater, HBO programming president Casey Bloys said the show is taking measures avoid revealing its ending. "I know in 'Game of Thrones,' the ending, they're going to shoot multiple versions so that nobody really know what happens," Bloys said in a recording of his Moravian College speech released and first reported by The Morning Call.

"You have to do that on a long show. Because when you're shooting something, people know," he added. "They're going to shoot multiple versions so that there's no real definitive answer until the end."

(Source)

The studio shoots and edits multiple endings, all of them plausible, so the production crew & cast do not know which will be used and cannot leak anything meaningful to the media.

Only the Director and an extremely few and carefully selected scriptwriters/producers know until the very last minute which ending will actually be used. That way it's a surprise to everyone.

Also, aside the already mentioned above, it could also can come down to budget or timing. If an ending is far too ambitious, it may exceed the remaining funds available and they have to either find additional funding or rewrite. If they are under pressure to complete before a date, a rewrite may be necessary and the already filmed ending might be scraped entirely.

Monty Python's Holy Grail did this, but instead of reshooting, they just took a left-turn and changed the ending to avoid shooting a costly and massive battle scene.

The movie was supposed to end in a large battle scene between King Arthur’s knights and the outraaageous French forces, but it quickly became apparent the production was out of money and couldn’t afford to shoot the scene. The solution? Just as the siege is about to begin, modern-day police enter the scene and shut down production, investigating the murder of famous historian ‘Frank’ earlier in the movie. Source

As such, there might be abandoned endings that never get full treatment, might have no CGI added or only simple CGI for placement.

Another reason might be to cut down the length of the film:

...the transmission tower for the plans was separate from the main base on Scarif. To transmit the plans, they had to escape and run along the beach and go up the tower. In cutting the film, it just felt too long. We had to find ways to compress the third act, which was quite long as it was. And one real, fast, brutal solution was to put the tower in the base, so they don’t have to run across the beach and do all of that stuff to get there. Source

I've seen it happen firsthand working on TV shows and movies, where they completely scrap the scenes, reshoot, or rewrite. I've got alternate versions that the world will never see floating in my brain, and if I watch the scene replacement, I can't help but to think of what it was.