I've seen many people on StackExchange and other forums mention that you can drop something as a free action, but I've never seen it sourced, or come across support for it in the Player's Handbook. The "Other Activity on Your Turn" section (p. 190) makes no mention of dropping items.
I've personally been ruling that dropping an item counts as an "interaction with the environment", but seeing how frequently I've seen people say otherwise I'm assuming I'm missing something.
The only indication we have within the rules that dropping things might be free is that dropping an object is not included in the (fairly extensive) list of example object interactions.
However, Jeremy Crawford has stated in an unofficial tweet from April 2015 that dropping something you're holding is not intended to require any action.
Dropping a weapon isn't specifically listed in the Player's Handbook, but I think it would be covered under "interacting with an object/the variety of flourishes" as described in the section "Other Activity on Your Turn":
Your turn can include a variety of flourishes that require neither your action nor your move.
You can communicate however you are able, through brief utterances and gestures, as you take 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.
You can do the flourishes and it says later you can ALSO interact for free. That would indicate the flourishes are not part of the interaction and therefore would not take the free item interaction allowing you to perform the flourish for free and then interact with an item for free as well.
Your turn can include a variety of flourishes that require neither your action nor your move. [...] You can also interact with one object or feature of the environment for free, during either your move or your action.
If that was not the case they would have said that you can do a flourish or interact with an object for free once per turn to indicate you could only do 1 of those each turn and you had to choose.
That being said, in the end, it is always the choice of the DM (I remember seeing somewhere in the official rules that basically stated the dm can choose to customize/change rules as they see fit).
If I was a DM and a player was throwing a fit because I did not allow them to do this all for free I would give in but have something happen as soon as they set the item down (perhaps a slight earthquake, an animal, an npc, or a trap but something) and this thing would either move the weapon away or steal it outright (possibly making the player lose the weapon for good).
Do that periodically and it would quickly teach them a lesson of why arguing with the DM may not always be the best idea.
The Player's Handbook does not provide a direct ruling for the mechanics of dropping a weapon. What the Player's Handbook (p. 190) does state is that you can interact with one object or feature for free and then goes on to provide a list of example interactions. Dropping an object is not included in the list of example interactions. As dropping an object is not referenced anywhere else in the rules it is reasonble to conclude that dropping an object fits best within the provided examples of free object interactions.
Some have argued that dropping an object can be considered a flourish and I would counter that by saying that a "flourish" would be best interpreted as something that conveys no mechanical advantage.
For example:
The only exception here might be when the act of dropping an object is intended purely as a flourish i.e. mic drop, stage exit.
Others have pointed to Jeremy Crawford's tweet on this topic that states "letting go of something requires no appreciable effort". I would consider this a RAI response rather than a RAW response, as it cannot be related to any RAW terminology and therefore should not be considered in a RAW interpretation.
In conclusion, the best RAW fit for dropping a weapon is to include this action under the "interact with one object or feature" list, however as the rules are not explicit this does require a DM ruling.
