I know that as the DM I can make up whatever I want and put the heroes of Faerûn wherever I choose in the story. What I am looking for is answers that are in other modules (published adventures) that I do not own yet.
For example: I think Drizzt shows up in Out of the Abyss, so I can hint as to where he is during the counsel meeting in Rise of Tiamat.
The reason I ask is that I know that my players will be asking where they (the heroes) are and also will want to know: why aren't they helping?
Specific heroes I am interesting in are: Elminster, Minsc, Farideh, Isteval, and the Grey Hand enforcers.
Best Answer
Well, let's see what we can find...
Timeline Background
The Tyranny of Dragons storyline kicks off in 1489 DR. Princes of the Apocalypse happens in 1491 DR. The pre-plot of Out of the Abyss got kicked off around 1485/1486, but the module itself doesn't specify when it actually happens - Chris Perkins said it happens 'right about the same time' as ToD and PotA.
Elminster
The Sage of Shadowdale does not lead a dull life. 1487 DR covered the events of the end of the Second Sundering, in which Elminster was intimately involved. He was quite busy preventing other people from ascending to the position of god of magic, and in restoring Mystra to her former power (recorded in the novel The Herald). In the year 1491, when PotA was happening, Elminster was in Waterdeep helping Larael Silverhand investigate the murder of several Masked Lords (recorded in the novel Death Masks).
What he was doing in the intervening years is unknown.
Minsc
Minsc is a character that only exists in video games, and video game tie-in comics (particularly Baldur's Gate and Neverwinter). In the Neverwinter game, events take place that do not match with the events taking place in published modules.
Minsc is one such case. The Neverwinter game's Module 6 (Elemental Evil) places Minsc working alongside the Emerald Enclave working against the Cults of Elemental Evil. Events in the game played out quite differently than they did in the published campaign.
Because he exists in a world that doesn't match up with the current published adventures, one cannot say what he was up to during the published adventures.
Farideh
We have no information whatsoever on the location or state of Farideh post 1486 DR. This is the time period in which the last book in the Brimstone Angels series was written, so we have no further information on what she is up to in the time period that the 5E published adventures are taking place.
Sir Isteval
Isteval turns up in the Tyranny of Dragons storyline. He is the representative of Daggerford for the Lord's Alliance at the Council of Waterdeep. It is emphasized that Isteval is a retired adventurer who now walks with a cane and has settled into a more political and advisory role... explaining his absence from actively helping out.
Gray Hands
Presumably, Force Grey is still around, as they are mentioned in the Waterdeep: Dragon Heist adventure. However, Force Grey's operations are mostly confined to Waterdeep and its surrounding areas. While this is not directly supported, it is likely that they are being retained by Waterdeep to tackle any threats that directly approach Waterdeep, while other Adventurers are supported and dispatched to deal with more distant threats.
One callout of note, however, is that an auxiliary member of Force Grey shows up in the Storm King's Thunder adventure.
Drizzt
The storyline of Drizzt's novels actually contradicts the events of Out of the Abyss. In this storyline...
Because his story contradicts the existing modules, it is hard to place him within them. However, this 'version' of the OOtA story takes place in 1486DR, technically prior to all the published campaigns. Which places him...
Or, perhaps, off on some other adventure that hasn't yet been written.
The Out-of-lore answer
Just so this is here...the obvious answer for 'where were they' is 'somewhere else, so the PCs can have the spotlight.' If Elminster and his allies showed up to try to solve the problem, there wouldn't be much left for the PCs to do.