Why are there no guns in Alien³?
I remember watching Aliens and being amazed with the variety of guns, war vehicles and ammunition shown. My expectation was that Alien³ would have at least as much action involving guns and in general follow death match style of the Aliens. However, if I am not mistaken we don't see many guns if not any.
Why was there such a drastic switch?
Why are there no guns or weapons in this movie according to IMDB:
The producers of Alien³ wanted each film to be different in tone and style. It is apparent that they did not want to make "Aliens 2" and made a conscious decision to shift away from the action genre. This also serves to remove the simple solution Aliens presented -- that the aliens can be killed quite easily if you have the guns to do so. The producers removed the guns to increase the threat to the characters. Sigourney Weaver also served as executive producer on the film and she is very anti-gun, though it is she herself who refuses/avoids handling firearms, not necessarily refusing their presence in a film in which she stars, as evidenced in Aliens and Alien: Resurrection.
From a personal point of view, having no guns gave the movie a whole other feeling, and I'm glad the crew opted to go this way. With guns, the aliens seem like a small threat, while without a gun a single alien can cause serious trouble.
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More answers regarding why are there no guns in Alien³?
Alexandre's answer gives a good out-of-universe explanation: my answer is the in-universe explanation.
Alien3 takes place on the prison planet Fiorina 161. This is a prison-work colony for violent offenders. From Wikipedia:
The pod crash-lands on Fiorina "Fury" 161, a foundry facility and penal colony inhabited by male inmates with histories of physical and sexual violence.
In the real world circa 2016, prisoners are barred from having weapons. I see no reason why it would make sense to give weapons to violent offenders currently serving a prison sentence even in the fictional future of this movie.
Why not provide weapons locked up so only the wardens have access?
This is an isolated, remote planet. It takes time to travel there: it appears that at the beginning of the movie a call would be sent out with the identity of the life boat, and it took the length of the movie (several days) for a response team to show up.
Looking at the prisoners, there are some big, tough dudes. Would you want to be the warden who can unlock the weapon room with those guys wandering around, when it would take several days for help to arrive? That is a recipe for mutiny, despite the themes of repentance and atonement in the movie. Having lethal weapons accessible to known violent criminals is too much of a temptation.
Besides, why do they need weapons to begin with? It is not every day that a bloodthirsty alien is on the loose, and a planet like that one does not appear to have much in the way of indigenous life that would be a threat to the colony.
All great answers and all containing interesting detail - however the truth of it is that they wanted and needed Ripley in the movie. Sigourney Weaver would only do it if there were no guns. So there were no guns.
The Brandywine producers did look at taking the story in another direction if they couldn't come to a deal with Weaver and there is a full script out there by William Gibson which focuses instead on Hicks and Bishop, which uses Communist-Bloc-As-Bad-Guys concepts, the first draft being very action-orientated, a la Aliens and the 2nd Draft condensing that down and dropping the bang-for-your-buck angle.
All-in-all, I think 10 writers produced different takes on the movie (including original director Vincent Ward's wooden planet (yep, wooden) and even a glass planet and even Fincher's finished movie was taken apart and reassembled by the studio before release.
Personally, I think the film is an underrated rough diamond - but maybe that's just me.
I do have some inside knowledge on this, as my writing-partner used to share his life with the film's script-reader.