Hi Jean,
I'm afraid that what are currently called "holograms" or "holographic displays", in fact usually have nothing whatsoever to do with actual holography, and in most cases do not provide a truly 3 dimensional reproduction of what they display. I apologize, but this common hyping and misrepresentation of the technology drives me to step up onto my digital soapbox.
The actual HOLOGRAPHIC 3D imaging process uses a laser to scan a 3D object - but the beam is split so that an interference pattern is created and recorded to a specific medium. In the early days of the technology, a laser was also needed to extract and reconstruct a 3D view of the image from the media. After "white light" holograms were developed (i.e. viewable without a laser), the 3D images could be mass-reproduced at relatively inexpensive cost and were printed on magazine covers, key rings, knickknacks, and other media. However, recording a hologram still involved technical expertise, minute calibrations, splitting a laser beam, and imaging a subject in a controlled, vibration-free environment. There are doubtless a number of current advances in the technology of which I am unaware, but that's a simplified description of true holography. If there is anyone out there with a better or more contemporary description, please feel free to elaborate!
What are commonly called "holographic presentations" these days are usually just a 21st century version of a stage illusion, originally patented in France around the time of the US Civil War - the "Pepper's Ghost". To see an excellent display of this technology, providing a truly 3D animated experience in use today, one needs only to visit the Haunted Mansion attraction at Disneyland in Anaheim. The giant ballroom featuring the dancing ghost figures is a large scale implementation of the original Pepper's Ghost (PG) illusion.
To see the same technology used in a museum environment, a quick detour to the Page Museum at the La Brea Tar Pits in Los Angeles, will provide an example of the "transformation" variation of the PG illusion (aka: woman to gorilla, as seen in carnival side shows). The Page Museum's display was in fact also created by the same "Disney Imagineer" who designed the ballroom illusion for the Haunted Mansion.
In the case of the Page Museum, they have a display showing a life-sized 3D model of a saber-toothed cat that dissolves back and forth between a 3D skeleton of the same animal, in the same position. The effect is achieved fairly simply, using two identical chambers, with two "mirrored" objects (cat and skeleton), a large sheet of glass, and controlled lighting that fades between the two chambers. The fact that physical 3D objects are used provides a truly 3D binocular experience.
What is usually displayed in "holographic theaters" such as the "Battle of Midway" presentation at the USS Midway Museum in San Diego, is a variation on this technique that uses multiple computer-controlled projectors to provide a synchronized presentation, with characters, objects, and elements that fade in and out, and due to differences in planar placement, offer a basic illusion of depth (although all of the projected images are 2D). Basically, one set of video displays are projected onto a screen similar to one in a regular theater. However a second set of displays are projected from above down to the "floor" of the theater, and are then reflected onto a sheet of transparent media set at 45° so that the images appear in the same orientation as the main projection, but on a different plane from it.
This technique was originally used to create "holographic" representations, integrated with live action performers, in a series of different concerts that included a band of animated cartoon characters, and a reappearance of the late Michael Jackson. One of the reasons this 2D imagery effectively appears as 3D is that the distance from the audience is great enough that the planar displacement overrides the usual binocular visual cues that indicate 3-dimensionality up close.
None the less, the PG technique may indeed be a solution for your exhibit. If up to three different states need to be displayed, you could potentially do it with 3 different 3D objects (I assume models of Mt. Shasta in different states). If you need to show more than 3 states, you'd probably need to use video projections, similar to what the Midway Museum uses, in different planar displacements to offer an illusion of 3 dimensionality.
Another approach, also used in the Disney's Haunted Mansion, might be to use "projection mapping". In the HM, it is used to animate the "singing marble busts" in the large graveyard scene. In recent years, fairly sophisticated software used for this sort of application has also become very affordable (although software is not an absolute necessity to create the effect). You might be able to build a physical 3D model of the mountain, or a cross-section of it, and then use video projection to alter its appearance over time or under different conditions. By projecting changing video onto what is essentially a white 3-dimensional screen (your physical model), you can offer a fairly convincing illusion of the physical object itself undergoing a change.
Just for inspiration, I'd also suggest that you do an on-line web search for Mt. St. Helens. There is a wonderful exhibit, simply created and placed on-site for visitors to the area to see how much of the mountain disintegrated. It essentially used an old film technique to provide a before image of the mountain printed onto a large sheet of glass, and positioned at the same location that the original photograph of the intact mountain was taken. By standing in front of the display, a visitor can easily see this image integrated with their live view of the mountain. But if one steps a few feet to the side, the current state of what remains is visible in stark contrast to the original image.
If you have any additional interest in developing a PG (essentially what the pyramidal display mentioned by Raymond is) or a projection map-based illusion, please feel free to contact me. I'd be happy to provide information on PGs from a presentation I made to an audience of "haunt enthusiasts" at a convention in Denver a few years ago. I'd also be happy to talk with you to get an better understanding of your specific needs, and possibly offer some more details or suggestions on how you might achieve your end goal at relatively minimal expense.