Neural network game players and incremental updates
The AlphaGo algorithm does not benefit from this kind of optimization.  It's hard to share an intuition for why without diving into the details of the algorithm, but at a coarse level: a single small change to the board can make a dramatic change to the strategic nature of the position, so it's not clear that one can reuse calculations effectively.  In particular, it sounds like you are assuming that AlphaGo does some computation separately for each square of the board, so we could avoid re-doing the same ones for squares that haven't changed.  However, that's not at all how AlphaGo works, so that kind of optimization is not applicable here.