I recently came across a theorem with the premise that "$I_1, \cdots, I_n$ are uncorrelated indicator random variables."
Nothing is said about the value of $n$, but I think it's safe to assume that, in general, it can be greater than 2.
On the other hand, I have not found any definition of "uncorrelatedness" for $n > 2$ random variables. More specifically, I have not found any definition of either the covariance or the correlation of more than 2 random variables.
What are these definitions?
I think what you're looking for is covariance matrix. Essentially, you view $(I_1,\dots,I_n)$ as a random vector in $\mathbb R^n$, then the covariance matrix, $C$, is symmetric with $(i,j)^{\mathrm{th}}$ entry $C_{i,j}=\mathrm{Cov}(I_i, I_j)$.
Saying that your random variables $I_1,\dots,I_n$ are uncorrelated then amounts to $C$ being diagonal, namely $$\mathrm{Cov}(I_i, I_j)=0,\quad\forall i\neq j.$$