The poison remains until piercing/slashing damage is done, or until it is washed off.
This appears to have been officially errata'd in 2015. Here is the relevant portion with the relevant sentence emphasized:
Poison (p. 257). [...]
The other three poison types have new descriptions:
[...]
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.
Thus, after the poison deals its effect once, it’s no longer potent.
The other poison types have new descriptions as well, for those interested.
I suspect the removal of a time limit is for speed of play/simplicity.
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.
Best Answer
From the rules:
Mundane ammunition
Magic ammunition
The poison doesn't matter, the arrow is destroyed when it hits, regardless.
Additionally, the poison rules say that the poison is consumed on a hit. The fact that the creature is immune to it doesn't matter.