Facing a large body of bodkin equipped archers in any armour was bad news.
I'm not aware of any known example of anyone being injured by an arrow penetrating plate armour. Bodkin arrows were very effective against other forms of armour, but not plate. The relatively long, thin head of a bodkin arrow is a
disadvantage against plate - the arrowhead bends. Gambeson, definitely. Mail, almost certainly. Plate, definitely not any main plate. Possibly the thinnest plate. I have seen accounts of French knights expressing concern about the possibility of arrows penetrating their helmets at the sides of their faces or through the holes in the visor that made breathing possible (but no evidence that either ever happened). But I'm not an expert. Maybe it happened.
The best test I've seen of period-correct arrows against period-correct plate armour (early 15th century in both cases) was organised by Tod Cutler and a small team of relevant specialists (armorer, bowyer, fletcher and archer). 160lb draw weight warbow with a 30 inch draw length at 25 metres range (very short range for that bow) against a realistically mounted breastplate, with a variety of arrows. Only a "platecutter" type arrowhead made from hardened steel (possible at the time, though no evidence exists of them) had any significant effect and even that wouldn't have caused any injury. It didn't penetrate the plate, let alone the mail and arming garment underneath. You'd certainly have felt it, though.
There is a common belief that archers often aimed for the face (when they weren't shooting for area effect), and a fair amount of evidence to suggest this was the case. Closed face helmets weren't particularly common, from memory, due to the disadvantage of vision loss.
Edit: Hugh Soar is a good author if you want to explore this. Robert Hardy's Longbow is (was? Been a while!) seen as definitive, but is hard going.
Oh aye, open face was a weak spot. Hell of a shot to hit at a distance with a medieval warbow and arrows, but definitely something to worry a lot about if you're facing thousands of archers each capable of loosing 6 arrows per minute at a sustained rate and with a well organised resupply chain on the battlefield.
Visored helmets became common, probably mostly as a result of that. An attempt to get a better balance between the advantages of open face and closed face helmets and generally a successful one but there are known examples of people being injured or killed by an arrow because they had their visor up when they thought they were out of range of archers but weren't. As well as the massive reduction in vision, there's also a significant reduction in hearing and the ability to breathe properly. So visor up until it needs to be down, especially if you're in a positon of command and thus really need to know what's going on. And are thus a prime target.
