A Wizard and a Knight clad in chainmail are happily walking through a forest. A wily Druid pops out of a tree and casts Heat Metal upon the Knight and he begins melting in his armor, screaming.

Rolling well on his Initiative, the Wizard decides to Polymorph the knight, who is very willing, to try and save him from a horrible, fiery death.

Heat Metal damages via physical contact:

Any creature in physical contact with the object takes 2d8 fire damage when you cast the spell.

Polymorph "melds" objects into the creature and common sense dictates that the object is gone. However, Polymorph is curiously worded:

The target's gear melds into the new form. The creature can't activate, use, wield, or otherwise benefit from any of its equipment.

Emphasis mine. Clearly, the description is saying that the user pretty much cannot use the object or be given any benefit, like a magic item. But can the object still be a detriment? After all, it still satisfies Heat Metal's requirement of being physically attached to the creature, and it is clearly not providing any benefit.

However, I can also understand the viewpoint that "the object" may not necessarily be an object anymore - it's part of the new creature. I do find it curious, though, that the stipulation of the creature not being able to benefit from or use the object is written in, rather than saying it simply doesn't exist for the duration.

Is the Knight still afflicted by Heat Metal while under Polymorph? Can the Druid still recast Heat Metal while the Knight is in his new form?

RAW, it is unclear, but everything indicates the Knight should be fine

This spell transforms a creature with at least 1 hit point that you can see within range into a new form.

The target's gear melds into the new form. The creature can't activate, use, wield, or otherwise benefit from any of its equipment.

It is unclear what melds means here. It does say it melds, not it disappears into the astral plane or whatever, so you could argue the metal is somehow imbued into the body of the new creature. However...

Choose a manufactured metal object, such as a metal weapon or a suit of heavy or medium metal armor, that you can see within range.

Even if the metal is considered to be somehow magically imbued into the new creature, it certainly is not a manufactured metal object any more, it's just metal blended into fur or whatever. So, first thing: We can no longer target that object.

If you already cast the spell, though, what happens? As you mentioned, if the metal is melded into the creature, you could interpret it as "being in physical contact", which is the only requisite for doing damage ("Any creature in physical contact with the object").

Continuing the reading of the spell, it says

If a creature is holding or wearing the object and takes the damage from it

Again, the creature is not holding nor wearing the object. So even if the Knight took the damage, he would not need to make the saving throw. Which is kinda awkward and inconsistent.

Thus, it is probably intended that the Knight is fine, but you could read that the metal being melded into the creature means it is still in physical contact with it.

A caveat is that the melded metal might not even be considered an object any more, so the whole wording of Heat Metal doesn't make sense any more.

Note: the question is tagged and asks for RAW, which is strictly interpreting as written. I'm just saying you can interpret it as still being in physical contact and thus still doing damage. I would never rule it this way, though.

I think the Knight is safe.

Polymorph essentially causes the target to swap stat blocks with a beast:

The target's statistics are replaced by the statistics of the chosen beast. It retains its alignment and personality.

Since no armor is mentioned in the stat block of a beast, heat metal can have no effect. The final sentence you quote is to make clear that you can't have a claymore-wielding bunny rabbit in plate armor.