Let me reply to your question with a related question.
My cleric worships Pelor, the god of goodness, healing, and the sun. What happens if my cleric wakes up at midnight and decides to draw runes in demon blood on his forehead and sacrifice a goat to Asmodeus?
I think the answer to this question is pretty clear. According to the Rules As Written, nothing bad happens. There are no rules about this situation, so the ritual has no effect and there are no consequences for your cleric.
However, that doesn't actually mean nothing bad happens. Your cleric has offended Pelor, and it is the DM's responsibility to tell you what consequences your cleric will suffer. The lack of rules just means the DM has to make something up.
In the same way, if a druid is forced to wear metal armor, it is the DM's responsibility to tell you what consequences that brings. As Dale pointed out, there are no rules for this situation -- but that doesn't mean nothing bad happens, it just means the DM makes something up.
As for what the DM should make up: the DM might start by looking at previous editions of D&D, such as this quote from D&D 3.5: "A druid who wears prohibited armor or carries a prohibited shield is unable to cast druid spells or use any of her supernatural or spell-like class abilities while doing so and for 24 hours thereafter."
I would also propose that the druid's animal companion tries to prevent him from equipping the armor, and while the druid is wearing the armor the animal companion does not recognize the druid (it treats the druid as a stranger).