Yes, false life and Heavy Armor Master work together.
Temporary HPs function exactly as normal HPs, except that they are designed to be lost first, before applying residual damage (if any) to your normal hit points. They are not to be mistaken for damage resistance, which is the ability to halve the damage taken in some situations, or damage reduction, which is the ability to reduce the damage taken by a fixed amount. All of them, however, should stack.
Player's Handbook (p.198)
When you have temporary hit points and take damage, the temporary hit points are lost first, and any leftover damage carries over to your normal hit points. For example, if you have 5 temporary hit points and take 7 damage, you lose the temporary hit points and then
take 2 damage.
If you possess some form of damage reduction, like the Heavy Armor Master feat, or some form of damage resistance, like the rage ability, you apply these effects first, just like you would when receiving damage normally (remember that, as per the rules found on p.197 of the PHB, damage resistance is the last modifier to be applied to damage taken). In effect, temporary HPs are a way to magically augment your ability to take punishment.
Contrast this situation with an abjurer's Arcane Ward. In this ability's description, the ward is described as a separate construct having it's own set of HPs, rather than giving you temporary HPs. As such, one could possess both an Arcane Ward AND temporary HPs (although temporary HPs from different sources still wouldn't stack), but the ward shouldn't, at least according to RAW, benefit from damage reduction or resistance effects. Indeed, it is not you taking the damage, at least until the ward breaks.
Yes, due to the wording of Heavy Armor Master
From the PHB, the feat is as listed:
While you are wearing heavy armor, bludgeoning, piercing, and slashing damage that you take from non magical weapons is reduced by 3.
If you separate these out into three separate statements (all are true, because they are engulfed within an "and" statement), they are as follows:
- Bludgeoning damage (from nonmagical weapons) is reduced by 3.
And
- Piercing damage (from nonmagical weapons) is reduced by 3.
And
- Slashing damage (from nonmagical weapons) is reduced by 3.
Then it stands to reason that each damage type would be reduced within that attack. This is my reasoning, as that is how and and or work in logical statements. If it was "Bludgeoning, Piercing, or Slashing" then it would only count once, no matter how many damage types were present.
Best Answer
Technically speaking, "damage [...] is reduced by 3" is neither Resistance nor Immunity, so it doesn't get overcome by a monk's fists.
That said, player-side effects like these don't often come into contact. It's certainly possible for two PCs to get into a fight where this matters, most often due to confusion, domination, etc., but ending up with this particular pair of abilities in conflict is the edgiest of edge cases. Monsters almost always use basic Resistance for this purpose. While I can't guarantee there's no enemy or spell anywhere that gives nonstandard damage reduction, I haven't seen it.
Because of that, as a DM, I can't see any balance issues with taking a broader reading that monk unarmed strikes just count as magic weapons any time it matters to an attack. I would not get too hung up on the exact text. It's never fun to tell your players "no, that doesn't work", especially when they seem at first glance to have a class ability that's specifically made for the situation at hand.
Normally, restrictive rules can be traced to one of two sources: a restriction to enforce game balance (such as, say, the rule about only being able to cast a cantrip at the same times as a bonus action spell), or a restriction to enforce the flavor of a specific ability (such as sneak attack being limited to small, quick weapons). Since this doesn't appear to do either, I'm left to wonder why it would be written this way. I suspect this is one of those situations where "helper text" accidentally made the ability more limited than it was meant to be. If the monk's ability simply said "the monk's fists count as magic weapons", that could immediately make your player ask "What does that mean? Why do I care?" (or worse, start thinking their unarmed strikes are innate +1 weapons), so I think they put in text that clarifies the benefit that magic unarmed strikes bring -- and it covers 98% of situations, but incidentally excludes a couple of weird corner cases that it maybe shouldn't.