How far has computer science moved past Knuth's TAoCP, if at all?
Matrix multiplication exponent has been improved and integer multiplication has recently reached speculated optimal complexity.

Perhaps Han`s sorting applies if correct (on which there was a post here).

Note Knuth`s book is his view. 

Modern view is richer. For example I doubt Knuth has spent much time on $NC$ algorithms for perfect matching or derandomization of primality or linear programming or counting algorithms.

So perhaps the algorithms he studies probably on whole not much more progress. However the analogy to mathematics that mathematics problems open in classical view does not mean mathematics has not moved forward holds in algorithms.