Does a poisoned weapon have to pierce the flesh to deal poison damage? For example, you can't pierce the flesh of lycanthropes with non-magical weapons, so I'm not quite sure the poison would be able to get into its system if attacked with a poisoned but non-magical weapon.
Depends on the type of poison. If it's an injury poison, unless the wielder has a magical weapon that CAN pierce the skin, then, no. But if it's a CONTACT poison, well, looking at their immunities, they are NOT immune to contact poisons. So if I had a non-magical blade loaded up with a contact poison, just touching them would be enough to get the effect of the poison.
Types of poison in D&D 5 include, with my notes in italics:
Contact: Contact poison can be smeared on an object and remains potent until it is touched or washed off. A creature that touches contact poison with exposed skin suffers its effects. Therefore they suffer the effects if they are touched, no need for injury.
Ingested: A creature must swallow an entire dose of ingested poison to suffer its effects. The dose can be delivered in food or a liquid. You may decide that a partial dose has a reduced effect, such as allowing advantage on the saving throw or dealing only half damage on a failed save. Hard to do in combat.
Inhaled: These poisons are powders or gases that take effect when inhaled. Blowing the powder or releasing the gas subjects creatures in a 5-foot cube to its effect. The resulting cloud dissipates immediately afterward. Holding one’s breath is ineffective against inhaled poisons, as they affect nasal membranes, tear ducts, and other parts of the body. Very effective against them, yes.
Injury: Injury poison can be applied to Weapons, ammunition, trap Components, and other Objects that deal piercing or slashing damage and remains potent until delivered through a wound or washed off. A creature that takes piercing or slashing damage from an object coated with the poison is exposed to its effects. Therefore, if no damage is taken, the poison cannot be delivered.
EDIT: However, in the comments, it's been pointed out that the lycanthrope immunity has less to do with not being able to pierce the skin at all and more to do with the fact that any non-magic/non-silver wound damage from those particular sources may close pretty much instantly. A DM could rule this way, but mechanics-wise, according to the rules, an injury poison would not work because it specifically states "a creature that takes piercing or slashing damage" and lycanthropes DO NOT take piercing or slashing damage from non-magic, non-silver weapons.
Damage Immunities are what matters
Lycanthropes have(MM, 208-211):
Damage Immunities: bludgeoning, piercing, and slashing damage from nonmagical weapons that aren't silvered.
There is no prevention of piercing the skin. Lycanthropes only gain complete immunity from any of the damage types listed above.
In the case of hitting with a poisoned weapon, the weapon's slashing/piercing damage would be zero, but they would still receive poison damage.
EXCEPT FOR INJURY POISONS
The DMG (pp 157-8) lists the various poison types. Injury poisons require:
A creature that takes slashing or piercing damage from a weapon or piece of ammunition coated with injury poison is exposed to it's effects.
The other types (Contact, Ingested, and Inhaled) do not require slashing/piercing damage first. Contact does require skin contact, though.
Depends on the poison type.
Excerpt - Poisons:
Contact: Contact poison can be smeared on an object and remains potent until it is touched or washed off. A creature that touches contact poison with exposed skin suffers its effects.
As long as the poison type is contact, your enemy just needs to touch it with exposed skin. The other types are Ingested, Injury, and Inhaled. If your poison is one of these types, then it gets a little trickier.
For lycanthropes, considering they have a good amount of exposed skin, using a Contact poison type will do the trick, if you are worried about not being able to pierce their skin or not.