Given a covariance matrix, it's simple to draw a venn diagram representing the independent and shared variance of two variables. How might one go about the same (at least, computing all pertinent areas; actual visualization not necessary) for the 3-variable case? Is this even possible?
A Venn diagram is more of an abstract way of illustrating events and probability space than a real tool.
If you have an experiment where events A and B can occur with some degree of dependence, then you can represent their joint probability distribution using a 2 by 2 table. The "upper left" corner indicates the probability that A and B are observed jointly. This is equivalent to the area of the overlap of the circles.
If there's a third event, C, then you can again represent their joint probability distribution using a 2 by 2 by 2 table. Again, the cell in the 1st table's upper left hand entry should represent the hyperarea of the intersection of probability spheres A, B, and C.
The tabular presentation of data and probability is more relevant to modern statistics whereas the Venn diagram is more of a learning tool for conceptualization.