The only ones I know so far are before clearing the first trial and if all wild Pokemon have certain status ailments. I don't actually play SM, so please don't question my curiosity.
Status Conditions - The only status conditions where it's unable to call for help are Asleep, Paralyzed, and Frozen.
Specific Pokemon - Specific Pokemon will not call for help whatsoever. Few of them are ones like Skarmory, Stufful and Bewear.
I only know one which is if there under a status effect