What is the minimum effective period over which one should try a new dietary plan, before reaching conclusions on its effectiveness?

Two weeks. But don't draw conclusions based on, for example, weight even if 'looking like you have lost weight' is actually your goal.

This question could be generalized to other health issues where self-experimentation is appropriate.

It could... but the more generalized the answer the less useful it is going to be. Specifics are kind of the important thing here!