Suppose $p(L)$ is a polynomial of negative powers of the lag operator, that is $$p(L) = \sum_{j=1}^\infty p_jL^{-j},$$ with real coefficients. What assumptions must $\{p_j\}_{j=1}^\infty$ satisfy in order for $$ \sum_{i=0}^\infty p(L)^i = \frac{1}{1-p(L)}? $$
My guess is that $\sum_{j=1}^\infty |p_j|$ must be less than $1$, but I can't quite figure out how to formalize this.
Yes, you need $$\|p(L)\|<1\tag1$$ for the operator series to converge after the theorem for Neumann series (geometric series in powers of some linear operator).
As $\|p(L)\|\le\sum_{i=0}^\infty |p_i|\,\|L^{-i}\|=\sum_{i=0}^\infty |p_i|$, it is sufficient to demand $$ \sum_{i=0}^\infty |p_i|<1.\tag2 $$ It is however not necessary, that is, there will be series that satisfy (1) without satisfying (2).