I think raw says the ongoing damage applies but just last week I told someone that since their shield of protection absorbed the damage of a duergar beard quill the poison didn't get through. This is as case of Rule of Cool to me.
Unless the immunity is a damage type not named (poison), in which case, only damage is negated. Alternatively, if the immunity is a condition or keyword not named 'poison' or 'gaze', the damage won't be negated.
So the rule you're thinking of exists for immunity only.
There are many substances that can poison you just by coming into contact with them. Being injured is not always required to introduce a toxin into your system.