Lack of assay sensitivity has different implications for trials intended to show a difference greater than zero between interventions (superiority trials) and trials intended to show non-inferiority. Non-inferiority trials attempt to rule out some margin of inferiority between a test and control intervention i.e. rule out that the test intervention is no worse than the control intervention by a chosen amount.
If a trial intended to demonstrate efficacy by showing superiority of a test intervention to control lacks assay sensitivity, it will fail to show that the test intervention is superior and will fail to lead to a conclusion of efficacy.
In contrast, if a trial intended to demonstrate efficacy by showing a test intervention is non-inferior to an active control lacks assay sensitivity, the trial may find an ineffective intervention to be non-inferior and could lead to an erroneous conclusion of efficacy. 
When two interventions within a trial are shown to have different efficacy (i.e., when one intervention is superior), that finding itself directly demonstrates that the trial had assay sensitivity (assuming the finding is not related to random or systematic error). In contrast, a trial that demonstrates non-inferiority between two interventions, or an unsuccessful superiority trial, generally does not contain such direct evidence of assay sensitivity. However, the idea that non-inferiority trials lack assay sensitivity has been disputed.