RAW there really is only one option to quickly take off armor: have someone help you.
Getting Into and Out of Armor
The time it takes to don or doff armor depends on the armor’s category.
Don. This is the time it takes to put on armor. You benefit from the armor's AC only if you take the full time to don the suit o f armor.
Doff. This is the time it takes to take off armor. If you have help, reduce this time by half.
The fastest a PC could shed heavy armor with help would be in 2 and a half minutes or 25 rounds of combat.
RAW options to avoid the damage
- Run away. As I interpret the spell you would still need to be in its range to use the bonus action attack. Running out of range would be difficult to achieve since the spell attack is a bonus action allowing the caster to use both their actions to follow you. However if allies impede the movement of the caster or your move speed is greater it could be accomplished in one round.
- Deal Damage. Every time the magic user takes damage while concentrating on a spell they are forced to make a CON save to keep that spell up. If everyone hits the wizard odds are pretty high that he will fail one of those CON saves.
Houserule option
Let the PC take it off with a standard action, but doing so damages the armor, preventing them from wearing it again until they have it repaired.
Musings on Heat metal
There is no requirement for a PC or NPC whose gear is affected by heat metal to react taking off their armor. The stipulation of the spell is to throw it if they can, donning and doffing seems to be a bit more than all that and as such I would interpret it to mean weapons should be thrown, but armor can be kept on. I know 2d8 bonus action damage sounds like a lot, but dice only damage has a way of only coming out to be around the average most of the time, while 9 damage a round without an attack roll is pretty nice, it is probably a better idea for the party to focus-fire on the wizard to end it rather than someone in heavy armor taking their armor off and seriously lowering their AC for the rest of the fight.
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.
Best Answer
No, it doesn't generally increase the weapon's damage
When you cast Heat Metal on an object you get the following effect, emphasis mine:
Anyone touching the object only actually takes damage from it once immediately when you cast the spell and thereafter when you use a bonus action to cause the damage again. Otherwise, touching the object (despite the heat) does not cause damage, so simply hitting someone with such a metal object wouldn't do any more damage than normal - you don't have to worry about accidentally giving your foes a boost by using the spell against them.
Action shenanigans!
The above said, there are a couple of ways I would allow you to get this kind of effect if you were deliberately trying to.
Firstly, if you readied the spell to be cast at the moment the target object was used to hit someone, I'd rule that both the wielder and their target were both in contact with the object at the time of casting, so both would take the damage.
Secondly, if you were to cast it on a weapon you were holding yourself, then on subsequent turns you could in principle use your bonus action to cause the extra damage exactly when you hit someone with it (since a bonus action without timing specified can be used whenever you like on your turn). That would get you one hit per round with some extra damage.
Obviously not a recommended strategy for use on yourself/friends unless you've got fire resistance or immunity.