Utterly nothing, and I'm not exaggerating. Various nations were withdrawing from the League as early as 1933 and then war broke out, League simply froze.

In the last meeting (December of 1939) Assembly delegated Secretary-General to run organization while session of Assembly and Council cannot be held.

It did not helped much, that Secretary-General Joseph Avenol, held rather pro-fascist views. Then war started, he fired most of LoN staff and left Geneva to join Pettain regime in 1940, August 31 (ironically, he ran back to Switzerland few years back, escaping German arrest). Sean Lester, Deputy Secretary General took over as final Secretary-General of the League after Avenol left. He, and few remaining staff members, spend the war in Palace of Nations representing organization that no longer functioned.

Then war started, LoN moved most of it branches outside Continental Europe in effort to keep them functioning, however most of them went dormant too. International Labor Organization is probably only branch that survived war and joined UN directly.

Others were reestablished under UN: International Refugee Organization replaced Office of the High Commissioner for Refugees under the Protection of the League, Permanent Court of International Justice was replaced by International Court of Justice and so on.