A great example is the climactic device in Lara Croft: Tomb Raider (2001). When assembled, it can be manipulated and modify the world, both in space and time. It looks like an orrery. A physically similar device from an earlier movie, Hudson Hawk (1991) plays a similar role. But instead of affecting the world of the film, it just changes lead to gold.
Another great example is the orrery of Prometheus (2012)
Two examples that miss: the crystal environments in The Fifth Element (1997) and The Dark Crystal (1982). Both of these affect the world, but as a tool not a model. And neither is mechanical.
A case where the machine does not resemble a traditional orrery is Thir13en Ghosts (2001). The device here is a machine of glass, masquerading as a house but actually an Ocularis Infernum, created by the devil and powered by ghosts. The moving parts give power over the future.
A similar case that does not qualify is La Chiesa (Demon Cathedral) (1989). The church is an elaborate machine that imprisons demons. When the machinery is activated, the demons are released and some humans trapped. The defeat of the demons is simply collapsing the building/machine.
Surely the most visually arresting example is in Ridley Scott’s Prometheus, especially when seen in 3D. What we have here is deliberately patterned after an orrery, in fact a specific painting of one. This passes our test in that it is a model of the world that goes beyond a simple representation of celestial bodies. The user directly manipulates it as a user interface, beyond just navigating through a star map.
The visual technique of holographic static snow is used in all of the computer generated images and user interfaces, even those that are human-generated. It is in a way the visual center of the film, the way Giger’s images were of Alien.
A reader suggests that the maze in Kubrick’s The Shining (1980) is a machine when people move through it. When Wendy and Danny move through it, they awake the hotel. Jack’s entering the Gold Room is the parallel. Well, that’s a clever view
Clocks
The most common examples are clocks. The number of films where clocks represent the world are numerous. Most just use the clock as a metaphor, but to pass our test, manipulation of the clockworks must somehow disrupt the world.
Examples where this is the case are:
The Hudsucker Proxy (1994) features a giant clock managed by an apparently all-seeing man, Moses. At a point near the end of the movie when the situation gets dire, he stops the clock,freezing time and allowing the dead to speak. Visually, many elements in the film’s world are based on clock-like elements: circles and lines.
For completeness, films where we have the metaphor but the clocks do not provide affordance are:
Scorcse’s Hugo (2011) deserves a special mention because it conflates special effects in movies, a mechanical man and a clock.
Fantastic Four (2005) the machine that recreates the storm?
Lemony Snicket's A Series of Unfortunate Events (2004): The orrery here is a clockworks in which the chronicler resides.
Hudsucker Proxy