There are some scientific works that deal with that topic. One way to realize a visualization support tool would be to determine data characteristics, interpretation objectives, practical application domain, cognitive abilities of the user etc. quantitatively and to use this information as input for a classification algorithm. Another way would be to use a rule-based expert-system operating on qualitative knowlege. However, it is a challenging task to model all these things. I'd like to learn about other approaches here!

As Benno said, it is an critical topic, if someone has an answer... I think that a Case Based Reasoning system could be the solution!

I think an additional problem would be that the audience itself is a variable. How people respond to certain forms of visualization depends at least partially on personal preference, as well as experience.

It also depends on the task the user wants to carry out.

I think the challenge is on getting to some reasonable metric. By way of example, take the interpretation objectives which Benno suggested (which I also  think is a good feature to include). Interpretation objectives has to be defined in a way in which it can be separable by the classifier given a training dataset for a supervised learning algorithm, a task which I think has nothing to do with ML but with human nature and the task at hand (not impossible but difficult).

Just a thought

In addition to what has been written above - computationally, I'd check the number of dimensions that the visualization can reflect. (since the visual presentation is basically flat and the data is multi-variable)

At the onset, I declare that I have never worked on visualization techniques. 

IMO interpretation of a visual  is largely subjective. E.g. Given a histogram looking at it gives an idea about the distribution of data. Computationally, one can only derive only distribution  information from the histogram. May be one can design a method to extract skewness, kurtosis, variance etc. from the histogram. My imagination limits me to think how would a computational method interpret the width of the class interval on the X-axis. Or if there are categorical  values on X-axis, how and what would a machine learn from it?

Further, a visual is already summarization of raw data. Processing a summary (histogram, pie chart etc.) for  further summarization may lead to serious loss of information.

In addition to my comment, here is an interesting link:

https://www.researchgate.net/profile/H_Schumann

I think it's worth having a look at the publications there (if you are concerned with more sophisticated diagrams/visualizations...).