Equipment and facilities needed for the demonstration
The equipment required is very simple - essentially just a standard hair dryer and the coil. It would also be helpful to have a small tray in which to retain the coil while it is being heated. There is also the issue of visibility. Unless the audience is very small, and can look down on the tray, they are unlikely to be able to see clearly what is happening. The best solution to this is to have a visualiser (camera sending its signal to a projector) available. Alternatively, an overhead projector can be used, although the visibility of the coil is unlikely to be as good as with a visualiser. Of course, it's also possible to just show a video of the demonstration, although there's no doubt that there are several advantages to having a "real" demonstration.
The key challenge is clearly that of obtaining a suitable coil. Actually, this is relatively easy. Wire of Ni-Ti (Nitinol) is obtainable commercially, and is not usually very expensive. It's probably best, if possible, to buy it as conventional wire - ie not trained into any shape - and to carry out the "training" in house. This will certainly be cheaper than buying a "trained" component. The training consists of constraining it into the shape concerned, heating it (to about 400-500°C) holding it for a few minutes and then quenching it to room temperature, probably just by removing it from the furnace and allowing it to cool in air. (Some trial and error may be needed, since the details depend on several factors, notably the composition and the diameter of the wire.) Various shapes could be produced, although a coiled spring is obviously an easy one. There are naturally some health and safety issues during the training operation - ie heating to ~500°C, and then cooling fairly quickly, but there are none of any significance during the demonstration itself.