Is there an algorithm or shortcut for continually updating and summing sub-sections of arrays?
The data structures you're looking for are quad trees (for linear time queries, logarithmic time updates, quadratic space) or the more complicated 2d interval trees (for logarithmic time queries, logarithmic time updates, $O(N^2 \log N)$ space).

I suspect you are trying to solve competitive programming problem, if so please state it in your question.