[RPG] What year is it in the Forgotten Realms during the Tomb of Annihilation adventure

The current year is not specified in the module anywhere and, for the purposes of the adventure, doesn't matter. Tomb of Annihilation happens more-or-less in a vacuum, no references are made within the module to events of other modules, and no novels have been released related to it. If there is a particular year that you like the idea of using, then go for it.

However, if you want to make sure you are in the right ballpark with regard to other published adventures, it would probably be somewhere after 1492DR. Thus far, every 5E storyline that has been dated has happened within the late 1480s (post-second-sundering) to early 1490s. This gives us a basic date range to consider

The second sundering, which is the event that moved us into 5E, ended in 1487 DR. This is our starting point.

The Tyranny of Dragons storyline takes place in 1489 DR.

Princes of the Apocalypse happens in 1491 DR

The plotting to set Out of the Abyss into motion got started around 1485/6 DR (according to a related novel: Archmage), but the actual events were said to happen right around the same time as ToD and PotA (per Chris Perkins)

The year for Storm King's Thunder is not set, but it makes references to Tyranny of Dragons as a past event. The phrasing implies this was relatively recent. Furthermore, there is a moment (p34) where...

Cultists from the Cult of Elemental Air turn up looking for allies. But the Elemental Cults are basically obliterated at the end of PotA...

So it seems that Storm King's Thunder is happening right about the same time as PotA.

The effects of the Soulmonger are very, very significant...so we know that ToA did NOT occur during the events of any of these prior adventures.

Thus, since the events of the world are fairly well booked from the second sundering up through 1491/2ish DR...if you want to slap a date on Tomb of Annihilation that makes sense with the other published adventures...then give it a date post-1492 DR