We do know that she doesn't follow the recommended dosages for things like cough medicine so she's probably abusing DXM for some kind of high. Given her history with ambien, I wouldn't be surprised if she's popping some sleeping pills too.
If I had to guess what's contributing to her getting pneumonia, my money would be on Brianna being a heavy smoker with a poor diet. Other than the Blue Apron meals, we don't ever hear about her eating anything other than junk food. Actually, based on the portion sizes we see her give Frank, I don't really know how much of that she even eats. She might still subsist on mostly Soylent and maybe a half-sized portion of the Blue Apron meals. You'd be amazed at how sick you can get when you aren't eating well.