Consequences of a quasi-polynomial time algorithm for the graph isomorphism problem
As far as I can tell, if you ask simply about the consequences of the mere fact (as a black box) that GI is in QP, I think the answer is very little. The one thing I can think of, which is not a theorem but a consequence for research directions, is to Group Isomorphism. Since GroupIso reduces to GI and we don't even know if GroupIso is in P, putting GroupIso into P can be seen as an important obstacle to putting GI into P (if you think the latter might be the case). 

However, since the trivial algorithm for GroupIso is $n^{\log n + O(1)}$, back when the complexity of GI was up at $2^{\tilde O(\sqrt{n})}$, we had a long way to go in improving the complexity of GI before GroupIso became an immediately relevant obstacle to putting GI into P. But if GI is in QP, then GroupIso becomes a much more relevant obstacle to further improvements in GI. (Of course, the exponent of the exponent in the quasi-polynomial is still a potentially relevant gap, but the gap becomes a lot smaller when GI is in QP.)