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In a particular adventure, the air in the lower levels is poisoned, causing 1d6 poison damage every hour:

From Hidden Shrine of Tamoachan in Tales from the Yawning Portal (p. 61):
"A creature takes 3 (1d6) poison damage every hour it spends in the gas. The gas is light, so it accumulates closer to the ceiling."

If a party take a long rest in such an environment, how do you calculate the cumulative effects of the poison air damage over time in combination with the restorative effects of the long rest?

The rules for a long rest state (PHB, p. 186):

At the end of a long rest, a character regains all hit points.

Does this mean, RAW, the characters take 8d6 damage during the long rest, but then 'wake up' at full HP? If after the nth hour, they drop to 0 HP and start rolling Death Saves, if they save, and make it to 8 hours, they wake up with full HP?

It seems like it shouldn't work that way, but I can't find any suggestion in the books that it would be otherwise.

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Frame challenge: don’t let them benefit from a long rest at all.

The rules for a long rest say:

A long rest is a period of extended downtime, at least 8 hours long, during which a character sleeps for at least 6 hours and performs no more than 2 hours of light activity, such as reading, talking, eating, or standing watch. If the rest is interrupted by a period of strenuous activity - at least 1 hour of walking, fighting, casting spells, or similar adventuring activity - the characters must begin the rest again to gain any benefit from it.

As a DM, I would rule that regularly taking damage is “similar adventuring activity”. 8d6 averages to 28 points of damage, which is enough to kill most non-adventurers. Actively having to deal with the effects of breathing toxic air is going to prevent you from getting any meaningful rest at all. At best, I might give you a short rest.

Alternatively, just let them take the long rest without issue.

Alternatively, it may work to just let them have the long rest. After all, the adventure states, as you quoted:

The gas is light, so it accumulates closer to the ceiling.

So it seems reasonable to rule that the gas wouldn't affect them if they were lying near the floor during the whole rest. It's up to you to determine ghow dangerous you want your dungeon to be.

Either way, worrying about damage during a long rest is not the way to go about this, in my estimation.

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    \$\begingroup\$ I would add - sleeping in a toxic environment is a pretty famous way to die. If they all go to sleep, they should be in serious danger. If they set a watch, the people who are awake should have an easier time noticing that breathing is getting painful in time to wake their friends up and go look for a safer place to sleep - or figure out if someone has a spell to protect them, at least. \$\endgroup\$ – gatherer818 Dec 28 '20 at 23:32
  • \$\begingroup\$ Alternatively, just let them rest normally, based on the last sentence quoted in the spoiler (maybe only in certain, appropriate locations). \$\endgroup\$ – Michał Politowski Dec 29 '20 at 12:01
  • \$\begingroup\$ @MichałPolitowski That might warrant its own answer \$\endgroup\$ – AncientSwordRage Dec 30 '20 at 13:51

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