I have created a fighter, who is using heavy armor and has the Heavy Armor Master feat.

The Heavy Armor Master feat (PHB, p. 167) states that:

While you are wearing heavy armor, bludgeoning, piercing, and slashing damage that you take from nonmagical weapons is reduced by 3.

This character was shot and hit, by a longbow loaded with a poison arrow. The attack did 2 piercing and 5 poison damage. Because of the HAM feat, the piercing was reduced to 0.

Because the arrow did not pierce (presumably because the armor absorbed the hit), would the character still be affected by the poison damage?

The shooter was a yuan-ti pureblood; presumably neither the longbow, arrow, or poison was "magical".

You take the poison damage regardless

The Yuan-ti Pureblood's shortbow attack states:

Hit: 4 (1d6 + 1) piercing damage plus 7 (2d6) poison damage.

Which means that you take the poison damage on a hit, regardless of whether you take piercing damage.

Crawford answered a similar question on Twitter:

Q: If the Battlemaster maneuver parry, reduce the weapon Drow damage to zero, the poison damage still work?

A: Drow poison in the DMG is delivered by piercing/slashing damage (0 dmg = 0 poison). Poison in the MM's drow is delivered by hitting.

Many drow have an attack with the exact same wording as the Yuan-ti Pureblood's shortbow attack, for example the Drow Elite Warrior's shortsword attack:

Hit: 7 (1d6 + 4) piercing damage plus 10 (3d6) poison damage.

Injury poison does nothing if the piercing damage is 0

There are poisons that behave the way you expected. These are injury poisons and are explained on page 257 of the Dungeon Master's Guide:

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.

However, these rules are meant for players and don't seem to apply to monsters. For example, the "drow poison in the DMG" which Crawford mentioned is an injury poison, which means piercing or slashing damage must occur for the poison to take effect, yet many drow have a hand crossbow attack that delivers an effect identical to the drow poison on a hit, regardless of the piercing damage.

Yes

Well, technically you don't get poisoned (which is a condition), but you take poison damage. You could argue that it doesn't make sense because the armor "absorbed the hit", but that is not what Heavy Armor Master implies.

Not getting hit is already determined by your armor class, the fact that the attack hit but the damage was then lowered a bit means the poison arrow did, in fact, get past your character's armor, your character is just tough enough to shrug off a bit of the piercing damage. The same can not be said for the poison damage.

Yes, Damage is damage - but you aren't poisoned.

The Yuan-Ti Pureblood can use a shortbow to attack. The stat block defines that attack as:

Shortbow. Ranged Weapon Attack: +3 to hit, range 80/320 ft., one target. Hit: 4 (1d6 + 1) piercing damage plus 7 (2d6) poison damage.

What you have is a successful hit causing both piercing damage and poison damage. The feat will reduce the piercing damage by 3HP, but it has no effect on the poison damage and that delivers as normal as the hit was still successful.

The Heavy Armor feat reduces nonmagical piercing damage and no more. It doesn't negate the hit.

The damage is just delivered on a successful hit via two different damage types and your feat helps reduce one of those types - it doesn't stop the hit so any remaining damage is still delivered.

See this related on question Does a zero damage attack still count as a hit?

Even if you wanted to tie the damage to a specific poison type and delivery mechanism, the fact that it did no damage to you from piercing doesn't mean that it didn't break skin. I don't think I'd say a small cut would be an equivalent to 1hp of damage, but I'm still cut and the poison could still be delivered.

You have gotten poison damage, but you aren't poisoned

I just want to clarify that getting poison damage isn't the same as being under the poisoned condition. That is a separate mechanic and is only active if something says it does that. Poison damage doesn't automatically impart the condition, it's just a damage type.

Poison is a Separate Damage Type

As Theik's answer pointed out, the poison damage here is not negated by the Damage Reduction of the HAM ability. That's what HAM gives you here: Damage Reduction 3 vs. bludgeoning, piercing, and slashing.

Even though the piercing damage is reduced to nothing by HAM, the attack still connected. You’ll find that explained in this question about zero-damage attacks.

You may also find this question about Damage Reduction useful.