I'm not aware of any drugs that he could have taken that would have this side effect.

Injections and drugs such as heroin can cause stroke, it's a common association actually. Commonly (depending where on the brain) a stroke will affect the primary motor cortex, or the cranial nerves, each causing facial drooping. Funilly enough it's not just the heroin itself, its the introduction of bacteria into the blood through injection too that can cause endocarditis.

Anything that mucks with blood pressure a lot too. So amphetamines and cocain, they are also associated with it. I'm not going to speculate on what happened to Bieber, I've no idea, but there is absolutely an association between facial drooping and some drugs.