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: 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 4: 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.
