Building on my answer here and the discussion in the comments, I think it's really up to the DM. The spell is written too ambiguously to rely on RAW. I can see it interpreted t̶h̶r̶e̶e four ways:
1. There is no disadvantage if the Constitution save is passed
This was my first interpretation when I saw the spell. The Constitution save determines whether the creature is required to drop it and the disadvantage is a consequence of not being able to fulfill that requirement, rather than a consequence of not dropping it. As previously stated (in case the other question is closed or deleted):
The spell refers to dropping it anytime the damage occurs and the Constitution saving throw is failed (emphasis added):
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. You cause the object to glow red-hot. Any creature in physical contact with the object takes 2d8 fire damage when you cast the spell. Until the spell ends, you can use a bonus action on each of your subsequent turns to cause this damage again.
If a creature is holding or wearing the object and takes the damage from it, 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.
As I understand it, if the Constitution save is failed, the creature must drop the object if possible. If the object cannot be dropped, then disadvantage comes into play. If the save is passed, the creature does not have to drop the object and does not suffer disadvantage.
This would be akin to the creature steeling itself to the heat and not being affected by it all.
2. There is disadvantage only if it's not dropped at anytime, regardless of the save
Per one interpretation, the spell only says "If it doesn't drop the object", so it could be valid that any dropping of the object eliminates disadvantage. This would be akin to the object still being too hot to handle effectively, regardless of whether the creature steeled itself to the heat (i.e., the creature can grip it, but it's still hot enough to cause distracting pain).
3. There is disadvantage only if it's not dropped immediately when failing the save
The third interpretation could be that the "drop" in the phrase "If it doesn't drop the object" means the drop from failing the save and only that drop. Even if the creature drops the item, the pain may linger for the turn, causing the disadvantage.
4. There is disadvantage only if it's not dropped immediately regardless of the save
This one feels the least likely to me if the answer to the linked question holds true, as a creature with a successful Constitution save would always suffer disadvantage (because it can't drop the item) while one that fails would have the opportunity to drop it. However it would be possible that the phrase "the creature must succeed on a Constitution saving throw or drop the object if it can" can be interpreted as "the creature must succeed on a Constitution saving throw or must drop the object if it can", which is another can of worms because then is the creature allowed to make a voluntary drop when it's not its turn?