What would be the complexity theoretic consequences of a quasi-polynomial time algorithm for the Graph Isomorphism problem?
More or less similar to the consequences of the deterministic polynomial time algorithm for primality testing, the deterministic polynomial time algorithm for linear programming, and the other case where practically efficient (randomised) algorithms (with rare pathological examples where the algorithm became inefficient) were know and in use for a long time. It confirms the conjecture that practical efficiency is a good indicator for the existence of deterministic theoretical algorithms overcoming the issues of the rare pathological examples.
Would a quasi-polynomial time algorithm for GI refute any famous conjectures in complexity theory?
No, the conjectures rather go to the opposite site, namely that GI is in P. Since GI is in NP, it won't be possilbe to refute this type of conjecture anytime soon.
Can we efficiently reduce Minimum Dominating Set in Tournaments problem to GI?
Minimum Dominating Set is not an isomorphism problem, hence there is no reason why it should be expected to be reducible to GI.
Is there any conjecture ruling out GI being hard for QP?
We don't even know how to reduce the string isomorphism problem to GI, and this is at least an isomorphism problem. Babai's proof showed that string isomorphism was in QP, so ... And what is being hard for QP even supposed to mean? Hard under polynomial time reductions?
From the introduction of On the Group and Color Isomorphism Problems by François Le Gall and David J. Rosenbaum
The complexity of isomorphism testing problems is worthy of study both because they are fundamental computational questions and also because many of them are not known to be in P, but nevertheless appear to be easier than the NP-complete problems. The most heavily studied of these is the graph isomorphism problem.
I believed I knew that all isomorphism testing problems of finite structures can be reduced to the graph isomorphism problem. Hence I believed that graph isomorphism was the "correct general" isomorphism problem to rule them all. The string isomorphism problem used in Babai's paper revealed that my belief was not fully justified, since it is still unknown whether the string isomorphism problem can be reduced to the graph isomorphism problem. Hence the generalized graph isomorphism problem $\mathsf{GI}^*$ and the generalized group isomorphism problem $\mathsf{GrI}^*$ are defined (in the above paper, but the authors rightly wonder why nobody did it before), which add the missing pieces from the string isomorphism problem. (And color isomorphism problem is just a different name for the string isomorphism problem. The name color automorphism problem goes back to the initial papers of Babai and Luks, the name string isomorphism occurs later in their paper on canonical labeling.)
Since Babai's algorithm was a quasi-polynomial time algorithm for the string isomorphism problem (i.e. for $\mathsf{GI}^*$), the consequence was that isomorphism testing for most types of finite structures should be expected to be quite doable. One application of such isomorphism testing is to list all different types of non-isomorphic structures with certain properties within a given range. Well, actually that application works much better with algorithms for canonisation (as opposed to mere isomorphism testing of two given structures), but the additional slowdown wouldn't change the principal polynomial or quasi-polynomial time bound for those problems.
Edit: This answer was given in the context of the retraction of Babai's result, before he announced a fix. It suggests that the slight generalization of the graph isomorphism problem suggested by the string isomorphism problem is the really important problem. The implicit expectation here is that any reasonable algorithm for the graph isomorphism problem will lead to a similar algorithm for the generalized graph isomorphism problem. The generalized problem is polynomial time equivalent to the set-stabilizer problem, the group intersection problem, the coset intersection problem, the set transporter problem, ... The idea behind this expectation is that the generalized problem will occur in the recursive part of any reasonable algorithm, so it has to be addressed anyway. (And it is quite possible the the generalized problem is polynomial time equivalent to graph isomorphism.)
Now Joshua Grochow's comments indicates that I wasn't successful in explaining the conceptual importance of the missing pieces from the string isomorphism problem. For infinite structures, it may be easier to appreciate that a valid isomorphism should not just preserve the given structure, but also belong to an appropriate category of functions (for example the category of continuous functions). For finite structures, the analogous phenomenon mostly occurs for quotient structures, where the appropriate category of functions should be compatible with the given quotients. The Johnson stuff is a typical example of such quotients, for example partition logic is working over the two element subsets of some base set. Also note that restricting the allowed category for the isomorphisms often makes the isomorphism testing problem easier, and that this might be the really crucial point for the importance of the generalized problem.
The problem with generalizations of the graph isomorphism problem is where to stop. Why not generalize so far as to encompass the permutation group isomorphism problem? This question is really hard, since many non-trivial results for graph isomorphism will probably carry over to permutation group isomorphism as well. But here it feels more reasonable to treat computational permutation group theory as a subject in its own right, even if it has indeed close connection to the graph isomorphism problem.
