All the data analysis-related texts I have read over the years recommend against using three-dimensional charts in almost all cases. To quote one of them, "Never use a 3-D chart when a two-dimensional chart will suffice."
That begs the question: in what instances are three-dimensional charts appropriate?
I'd say it is sometimes appropriate when you want to use that dimension to convey an additional dimension of information i.e. you have X, Y & Z axes. This can be especially useful when you have the ability to rotate/animate the graph.
However, there are other ways of communicating extra dimensions. It is often more effective to use colour as a third dimension. And it is sometimes useful to create cutpoints in the third dimension and plot panels for each of the data subsets.
The standard usage of making a bar or pie chart 3D is on the other hand, utterly useless, because the extra dimension conveys nothing.
This question brings to mind Minard's map of Napoleon's Russian campaign, which has been called "the best statistical graph ever drawn" by E. Tufte. It manages to represent 5 dimensions (2D cartographic, army size, time and temperature) on a single 2D chart; quite a feat. And it is also quite easy to read, and esthetically very pleasing.
Having said this, and at the risk of stating the obvious, a 3D chart is very appropriate when your data is 3-dimensional (3 and exactly 3 continuous variables). One example would be a response surface plot, as the result of a DOE, optimizing an outcome, as a function of 2 independent variables. The response surface is the most natural way to display the results.
Still at the risk of stating the obvious, when your data is 2D (only 2 variables) a 3D plot is useless (as the 3rd dimension is just "cosmetic"). And if your data is n-dimensional ($n>3$), then a 3D plot of a "cross-section" is not easily interpretable.
Now, can you represent 3D or higher dimension data on a 2D plot? Yes, and often well. Minard did it for 5D, and a bubble plot can do that for 4D data (3 continuous variables, 1 categorical data). But on the other hand I would not quite consider isolines as equivalent to a good 3D plot, particularly when one can rotate/manipulate it (because isolinesit bin the 3rd dimension, thus losing some information; but in many contexts, e.g. hiking, it is basically just as good).
Now, as the quote mentionned by the OP basically amounts to "never use a bulldozer when a shovel will suffice", I can not find fault with that statement. But that does not quite translate to "recommending against using bulldozers in almost all cases".
At the risk of vacuity, I would say that three-dimensional displays are justified if and only if they are clear, not that one- or two-dimensional displays are different in that respect.
There are at least two cases.
For two variables $x$ and $y$ there is at most one value of a third variable $z$. Such data may resemble that of a topographic surface (e.g. on land or under water). Sometimes one or more perspective views of such surface data may be helpful, but the risk is that parts of the surface are occluded by others, just as when you view a landscape -- even from on high -- nearby hills and mountains will hide whatever lies behind (although that is what "behind" means).
Often the best representation is a contour map. Contours are lines of equal height either above or below some reference level, e.g. sea level. Contours in general are sometimes known as isopleths or isarithms, although there is a wealth of special names, ranging from well known, such as isotherms and isobars (which may be familiar from meteorology), to obscure, such as isonoetic lines (lines of equal IQ, if I recall correctly).
A physical model is often evocative (even fascinating) for longer-term display in some exhibition, but far from easy to produce for most routine analyses.
A point cloud defined by $x, y, z$ triples. Sometimes a perspective view helps and sometimes it doesn't.
This case is very common, but most successful attempts to convey pattern or structure entail some projection to two dimensions, such as that of two principal components from principal component analysis, or similar constructs from other multivariate methods.
But what is two-dimensional or three-dimensional? The case of a contour map has already been mentioned. We can add choropleth maps (Tukey's homely term patch map deserves more use), in which areas (e.g. political or administrative units) are shaded according to values for each area. Are these displays three-dimensional?
The difficulty of going beyond two dimensions is underlined by how many clever ideas to show more dimensions indirectly turn out to be problematic in practice. Top of my personal list would be Chernoff faces, ingenious but useless!
Here is a practical example of a 3D view. This is a screenshot of a tool called 'zview' which is used to visualise and edit colour transforms. The RGB colours of the spheres represent the RGB exposures of the original internegative film. The XYZ coordinates represent the colour we see with respect to the open gate white using a xenon lamp. The red arrows show the shift in colour with respect to white if we use a different lamp in the projector.
In 'zview' you can rotate the view by dragging the mouse. This gives a much better impression of the 3D shape than this static image. I normally use this to inspect transforms as represented by lists of data with 3 independent values and 3 dependent values per point. In this example, we show the shift between two projector lamps, which would make 3 independent values and 6 dependent values per point.
With an ideal audience, it is certainly appropriate to use 3D representation if you have a point to communicate about varying relationships among three or more independent attributes (continuous OR NOT). However, many institutional cultures have conditioned people to believe that 3-D representations are inherently bad. This culture literally includes complaining that 3D representations are too hard to comprehend, even though those complaining may have PhD's. The cultural values of discouraging such use (sometimes prohibiting 3D!) are understandably influenced by scenarios in which 3-D diagrams have been inappropriately used or poorly designed. Regardless, you may find resistance to use of 3D images even when the audience is seeking to understand a 3D relstionship. You must know your audience's culture regarding 3D. Just one or two people who are resist(well-designed) 3D images can ruin your attempt to communicate to your audience.
