Necklace of Fireballs (DMG p.182)
You can use an action to detach a bead and throw it up to 60 feet away. When it reaches the end of its trajectory, the bead detonates as a 3rd-level fireball spell (save DC 15)
RAW the bead has not been thrown to be able to reach the end of a trajectory so nothing happens. This has already been said.
However the "rule of cool", the expectations of the consequence of such a stupid act suggest that this should probably not be so anti-climactic. There are many examples in popular culture fiction of someone swallowing something that will go bang (XMen First class is the one that springs to mind straight away) and in this case the DM should feel free to house rule that it does indeed count as having been triggered, and the end of it's trajectory is the stomach of the Teifling (hereafter known as the idiot ;) ).
As to the effects of this a fireball is constrained by barriers, even if only the idiot's chest, so it will only effect the idiot themselves. It is clear that Dexterity saves against swallowed fireballs should automatically fail, much as if the target was paralysed as far as this effect was concerned and they take the full 8d6 damage, doing damage in the range 8 to 48 with an average of 28. As they have fire resistance then this will be halved to a range of 4 to 24 with an average of 14. There is a good chance a reasonable level character will survive this.
However there is a rules option you may want to consider given the internal damage nature of this.
Lingering Injuries (DMG p.272)
It's up to you to decide when to check for a lingering injury. A creature might sustain a lingering injury under the following circumstances:
- When it takes a critical hit
- When it drops to 0 hit points but isn't killed outright
- When it fails a death saving throw by 5 or more
These are just suggestions of when a DM might decide to apply a lingering injury. I think judging that eating a fireball should count as a condition for sustaining a lingering injury is very reasonable.
And then there is the Massive Damage optional rule which may apply also, depending on how much damage was done:
Massive Damage (DMG p.273)
When a creature takes damage from a single source equal to or greater than half its hit point maximum, it must succeed on a DC 15 Constitution saving throw
or suffer a random effect determined by a roll on the System Shock table.
Obviously whether you can apply this depends on the hit point total of the idiot.