I'll take a stab at answering my own question.

The missing “something” in the edited version of my question appears to be the mutual
information of one or more events, denoted $I(A,B,C,...)$. More precisely,
it appears to be the quantity $e^{-I(A,B,C,...)}$. It would be good if
someone knowledgeable about information theory could weigh in here—I haven't
found a definition in the literature of the mutual information of more than
two events, but the one I give below seems to be the “right” one.

In the information theory textbooks I looked at, the information of a single
event E is defined as I(E) = −log P(E). The mutual information of two
events, E and F, is defined as
$I(E,F) = -\log \frac{P(E) P(F)}{P(E\cap F)}$,
and is a measure of the degree to which E and F fail to be independent. That is, $I(E,F)$
is zero if E and F are independent, positive if they are positively correlated, and negative if
they are negatively correlated.

The appropriate generalization to three events seems (following the
suggestion of Kenny Easwaran) to be
$\displaystyle I(E,F,G) = -\log \frac{P(E) P(F) P(G) P(E \cap F \cap G)}{P(E\cap F) P(E \cap G) P(F \cap G)}$
which makes some sense as a measure of the failure of independence since,
in order for E, F, and G to be independent, it is required, not only that
P(A ∩ B) = Pr(A) P(B) for all pairs of events, but also that
P(E ∩ F ∩ G) = P(E) P(F) P(G).
(Does anyone know if the definition of I(E,F,G) given above is standard?)

Now define C({A,B,C,...}) = P(A ∩ B ∩ C ∩ ...). The appropriate
generalization of the mutual information to an arbitrary number of events
seems to be
I(E,F,G,...) = −log [Π_{S} C(S) / Π_{T} C(T)]
where the product over S runs over all subsets of {E,F,G,...} of odd
cardinality and the product over T runs over all subsets of
{E,F,G,...} of even cardinality. (Again, does anyone know if this definition
is standard?)

With these definitions, we get the inclusion-exclusion-like rule,

−log P(E ∩ F ∩ G ∩ ...) = I(E) + I(F) + I(G) + ...
− I(E,F) − I(E,G) − ... + I(E,F,G) + ... − ...

This can be proved by a counting argument identical to the one used to
prove the principle of inclusion-exclusion. Negating and exponentiating
both sides produces an identity of the desired form. It would be nice
to also find a Möbius-inversion style proof.

As to whether this is “useful” in the same sense that the principle
of inclusion-exclusion is useful, I can't say. The definition of I(E,F,G,...)
is itself an inclusion-exclusion-like rule, so it's tautological that when you
invert it to find −log P(E ∩ F ∩ G ∩ ...) you will get an
inclusion-exclusion-like rule. I suppose the significance of all this depends
on how fundamental the mutual information is.

*Addendum:*
The book *Elements of Information Theory* by Cover and Thomas mentions the problem of defining the mutual information
of three random variables in Problem 2.25. The mutual information of random variables is related to but somewhat different from the mutual information of events since, for example, I(X,Y,Z) involves taking the expectation over all events (X=x, Y=y, Z=z). The problem notes that, in contrast with the two random variable case, the mutual information of three random variables is not a non-negative quantity in general. This perhaps explains why not much theory has been developed around it. Interestingly, however, the mutual information of three random variables can be expressed in terms of the entropy of one, two, or three random variables via the inclusion-exclusion principle.