Thought of while contemplating this question on whether you can drop prone off your turn, I am wondering if you can drop a weapon/item.

This Q/A claims that dropping a weapon is a free action, but I am not sure if this means it is something that can be done when it is not your own turn.

This Q/A says that you cannot speak off your turn because it is explicitly limited so, and this Q/A makes a similar argument for why you do not get an object interaction during Readied movement.

Whatever the ruling is, it would also affect the heat metal spell which states:

the creature must succeed on a Constitution saving throw or drop the object if it can. If it doesn't drop the object, it has disadvantage on Attack rolls and Ability Checks until the start of your next turn.

So succeeding on the Constitution save which is not optional could potentially prevent you from dropping the item.

Can you drop a weapon/item when it is not your turn?

The question "Can you drop a weapon/item when it is not your turn?" is confusing primarily caused by the word "can" (including in the heat metal spell) so we have to address the portions separately.

1. You can be forced to drop an object not on your turn, such as by the Battle Master's disarming attack ability.

Disarming Attack When you hit a creature with a weapon attack, you can expend one superiority die to attempt to disarm the target, forcing it to drop one item of your choice that it’s holding... On a failed save, it drops the object you choose. The object lands at its feet.

Effects that take place on the target's turn, rather than immediately (such as from the Command spell), explicitly say so.

2. You cannot, of your own volition, interact with objects or take free actions not on your turn, including dropping an item.

You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack. (PHB)

Allowing free interactions not on your turn is not described in the Combat section of the PHB, which is assumed to be an exhaustive list of options. Additionally, it would cause severe conflicts with both action economy and initiative, among other problems.

3. The "can" in the heat metal is referring to physical ability.

Your original question is defining "can" as "ability within D&D mechanics." The heat metal spell is defining it as "physical ability to do so." A sword, should you fail the constitution save, can be dropped. Items that have a doff time such as a shield (1 Action) or armor (1-5 minutes) physically cannot be dropped, regardless of the save outcome. (PHB Ch. 5, Getting Into and Out of Armor)

This was harder to research and support than I'd first thought! I'd expect house rules to vary a lot on this one. I may be misunderstanding what's being asked, but I believe the answer to this is

RAW, you cannot choose to drop a weapon when it is not your turn.

Reason 1: You generally get to do things like this only on your turn.

The relevant section from the Player's Handbook:

Other Activity on Your Turn

[...] You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.

This isn't as on-point as I'd thought before looking it up, as it doesn't talk explicitly about dropping things. But I would argue that dropping a weapon is similar to interacting with an object, and as such would be bound by the passage italicized above: during either your move or your action.

The list of example actions that can be done in tandem with movement and action contains many options that are about as strenuous as dropping something, so I don't see a reason that dropping things would be a special exception to the condition that these things take place during your turn.

Reason 2: There are conditions in which you can act out-of-turn, but these are specific and either rely on readying an action for use at a specific, later time or explicitly using a reaction mechanic or other specified ability.

You can ready an action based on some triggering event, which may execute after your proper spot in initiative order, but you still have to use your actual turn to prepare that action.

Combat rounds play out "simultaneously", with your turn defining when you get to choose to do things. Taking action out-of-turn without using these mechanics cuts against every other mechanic regarding order of actions that I can think of. Without a specific reference stating otherwise, I see no reason that dropping an item would be an exception to this.

So if you readied an action of "I'll drop my weapon" on the condition of "my weapon becomes hot" (or any other condition), that would work even when it's not your turn. But reactions are not available in the moment. The action has been readied, and is available as a reaction, or it hasn't, and isn't.

Reason 3: Effects exist which cause characters to drop weapons, and there is no suggestion that this is not immediate nor discussion of consequences of a delay.

Here I'm thinking specifically of disarm, which states (Dungeon Master's Guide):

[...] If the attacker wins the contest, the attack causes no damage or other ill effect, but the defender drops the item.

There is no mention or suggestion of any sort of delay, and it seems mechanically odd that the disarm action would be successful but that the weapon would only leave the character's hand at the start of their next turn (potentially allowing it to be used for a reaction, etc. in the meantime). There is no mention of an option to choose to drop the weapon after winning the contest.

Reason 4: Spells do what they say the do.

It's possible that the "if it can" wording could be a reference to heat metal being cast as a reaction during the target's turn, so I suppose we can't rule that out completely. But other spells exist to impose disadvantage, and this would be a very roundabout way to describe only that same effect.

And since the saving throw isn't optional, and the target can't choose to fail it, it seems odd to "cheat" the consequence of a high CON saving throw by taking a free action at a time when you ordinarily couldn't do so.

The wording is poor with "if it can [drop the weapon], but I really think that the intent of that phrase applies to a case where the weapon is freely held versus being physically attached in a way that makes sudden dropping impossible. It seems less likely to me that that qualification describes arbitrary meta information like turn order, but as above, it's at least a potentially plausible reading.

Reason 5: A deliberate action isn't what Constitution represents in a situation like this one.

I'm loath to fall back on verisimilitude when talking about rules, but as the wording is a bit ambiguous and the rules don't address this exact situation I think that it's worth considering. Constitution definitely applies to many voluntary actions, like continuous marching or holding your breath.

But it also applies to involuntary ones, like resisting poison. You don't get to choose whether or not you want to suffer the effects of poison after being poisoned, you either succumb to it or you're capable of withstanding it. A case like heat metal strikes me as similar: if your Constitution fails (because it's low or due to an unlucky roll) then you are incapable of holding onto the weapon and reflexively let go of it. If your Constitution save is better, you notice the heat but are able to endure it (if imperfectly).

The saving throw doesn't give you an abrupt mini-turn, nor does it change initiative order. It instead determines if you are capable of holding onto the weapon. Making a deliberate choice to drop it is governed like any other deliberate choice: you choose what to do on your turn and then do it, or you use one of the existing mechanics which explicitly allow for out-of-turn action.

Yes, you can drop a weapon when it is not your own turn

Crawford, though now unofficial, has stated:

The intent is that letting go of something requires no appreciable effort. But picking it up does...

We, in fact, do know exactly how much effort is required to pick something up as this is specifically called out under "Other Activity on Your Turn":

You can also interact with one object or feature of the environment for free, during either your move or your action...

INTERACTING WITH OBJECTS AROUND YOU

Here are a few examples of the sorts of thing you can do in tandem with your movement and action:

...

pick up a dropped axe...

So if picking up a dropped axe requires your object interaction and dropping something is, RAI, supposed to take even less effort, all I can think of is that it takes no action whatsoever.

Whether or not dropping something takes an object interaction, RAW, is complicated but as this answer points out, it is likely free as it is not listed in the fairly extensive list of interactions with objects. (In addition, both closing and opening a door is listed, whereas only picking up an axe is listed and not dropping one).

So dropping something takes no action, and as there is no text saying that it can only be done on your turn, like there is with speaking, it can be done off your turn as well.

There are two things I know of in the rules that are similar to this:

Under "Grappling" it states:

The condition specifies the things that end it, and you can release the target whenever you like (no action required).

This means you can release the grapple even when it is not your own turn, as it requires no action.

Similarly under "Concentration" it states:

You can end concentration at any time (no action required).

And as this Q/A agrees, this means you can end it at literally any time, even when it is not your own turn.

I believe dropping something is a free action, similar to ending a grapple or stopping concentrating on a spell, and thus it would act similarly to those and can be done as a free action at any time.

Further evidence that dropping an item is a completely free action:

There are also a great many questions and answers such as this one which assume dropping a weapon takes no effort and have used that in their builds.

There is also this Q/A which says you can remove one hand from a weapon as a completely free action and this Q/A which makes a good argument for why removing a hand from a weapon is a free action.

Because dropping a weapon is a completely free action, like the other two cases above, you can drop something at any time, even when it is not your turn.