I don't know about every type of polearm but we can speak specifically to pikes which dominated late-medieval and early renaissance warfare as there are quite a number of discussions about this topic w.r.t. to those. In our medieval arms and armor AMA a few months back we had a question about how medieval armor would stack up against bows which has some great citations with it and the sentiment it presents has been pretty commonly repeated throughout every discussion on this topic I have encountered. The comment.

How well did armor hold up against powerful bows like longbows or the recurved bows commonly used by horse archers?

It held up reasonably well. The thickness of this armour varied across the body, breast plate and helmet were thickest while arm and leg armour would be thinner. The bulk of modern experimental archaeology, a field not without its problems, suggests that longbows and similar weapons were able to penetrate thinner armour with some force but were largely ineffective against breast plates and helmets. This meant that a well placed longbow arrow, and it should be of course noted that hitting a moving target straight on with an arrow was a tricky thing, could maim or injure a knight in full plate but short of a lucky hit on an artery would not kill. A lot of these tests don't account for undergarments worn beneath the armour either so some of the effective injuring potential may be overstated. The tests are also of arrows fired straight on at the target rather than in an arc and so are at a relatively short range. At a greater distance the armour would likely hold up better.

Basically the pretty universal sentiment I've come across is that armor alone basically rendered arrows not nearly as deadly as you paint them as. /u/Rittermeister (hopefully this will summon him!), our resident Medieval Historian extraordinaire, stated in this post that, quote,

The basic, short, self bow (self meaning it's made from one piece of wood, not multiple layers glued together as in composite bows), such as would be useful for hunting, is virtually useless against an armored infantryman. We see this in the chronicles of the Crusades, when both Christian and Muslim accounts report the complete inability of their bows to pierce mail backed with padding. They are simply not launching the projectile hard enough to actually get through.

He goes on to say (and this is another commonly stated thing) that advances in armor building began to make even the infamous longbows nearly ineffective against armor.

Here is also a great post about why European pikemen did not use shields. The relevant bits:

I think a better answer would note that shields in general were in decline in European warfare at this time, or at least radical transformation when it came to their traditional function. First, the introduction of steel plate armor made shields increasingly redundant, as anything capable of getting through your breastplate would probably pierce a shield. Men-at-arms began to use two-handed weapons almost exclusively, and shields like the buckler became more associated with light armor.

I also liked a reply further below which stated something I believe the top post was also trying to convey (but did it a bit better):

The Swiss pike blocks evolved around pressures of facing heavy shock cavalry charges instead of massed archery. Since the main threat to the formation is a heavily armored charging knight, the Swiss formations were built around repelling that threat. The pike and the mobility and discipline of the men wielding that pike was a weapon system that suited its time period.

I expect these answers to also extend to other 'two handed' weaponry though. Improvements in armor manufacturing, the dominating force in medieval Europe being the heavily armored shock cavalry, and a penchant for aggressive movements basically made the shield less useful than before. That's not to say shields disappeared though from the medieval battlefield (as more medieval inclined flairs will certainly come in to say later on) but as said above: the later the Medieval era went (late 1200's - 1400's) the more redundant a shield became especially in the face of who was their largest threat; cavalry.

Prior to the widespread adoption of plate armor for even regular footsoldiers, static, unprotected formations of pikemen did get shredded by archers. Scottish formations were repeatedly cut to bits by English archery, like at the Battles of Falkirk and Dupplin Moore. However, the key words there are "static" and "unprotected." In an ideal situation, the pike block would be protected by allied cavalry and screened by their own skirmishers. When under competent command, a properly trained pike formation, like the Swiss, would attempt to close with the enemy as quickly as possible to avoid being cut apart by archery or gunfire. This was successful on some occasions (Murten, Nancy) and led to terrible casualties on others (Cerignola, Biocca, Marignano, Pavia).