What is the official position of the Jehovah's Witnesses in regards to hell, specifically the doctrine of annihilation which says that people will not go on existing, that eventually God will just extinguish some/all souls rather than leaving them in a state of eternal punishment.
Short answer: Jehovah's Witnesses do not believe in Hell.
Longer answer: Technically, Jehovah's Witnesses do believe in Hell, but (a) their beliefs about Hell are quite unusual, and (b) they generally don't use the name "Hell". (So it's simpler to say they don't believe in Hell.)
Even longer answer: The Biblical words usually translated as "Hell" are the Hebrew She'ol and the Greek Hades. Witnesses use these words (more often She'ol, in my experience, but the two can be used interchangeably), but they can occasionally use the English word Hell to describe the same concept. She'ol is not a place, physical or otherwise, but a condition: it is the state of non-existence. And everyone, when they die, simply ceases to exist (see next paragraph for an exception). This applies even to Jesus: when he died, he died. And then was raised after three days. So too the other resurrections in scripture: they were brought back from nothingness.
This applied also to the 144,000 anointed ones. They died. And then they were resurrected to Heaven in 1914. Any anointed who have died since 1914 are resurrected straight to heaven, in a "twinkling of an eye". So that's the exception.
In the new world order, many people from She'ol (Hell) will be resurrected. Those who rebel will, at the end, be thrown into the lake of fire. This indicates complete annihilation: the same sort of non-existence as in She'ol, but without any chance of future resurrection.
Another term sometimes confused with Hell is Tartarus, which I believe is used only once in the Bible. The Witnesses believe that's a completely different thing: it's the condition of the Devil and his demons cast out of Heaven.
Jehovah's Witnesses do not believe that there is a soul that survives the death of the body. We believe that, at death, a person ceases to exist, since the body simply returns to the "dust". (Genesis 3:17-19) This makes sense if you think about people who have brain damage (a damaged body): their consciousness is affected by this. Our consciousness is not an independent system from our body that mysteriously survives the death of the body. God created the first man and woman to have a physical body formed from the "dust". (Genesis 2:7) This is different from the angels he created which are spirits. (Ezekiel 3:12) God is also a spirit. (John 4:24)
Jehovah's Witnesses don't believe in hellfire as it would not represent divine justice from a loving God. (1 John 4:8) He wants us to serve him out of love, not fear of torture. (2 Corinthians 9:7) Some who die will receive a resurrection, but some will not. If God sees that someone has already shown that there is no chance they will conform to his righteous standards, he will simply leave them in a state of non-existence. (Matthew 5:30) They have already paid for their sins with death, and he has no desire to torture them. Those whom God chooses to preserve in his memory will at a future time receive a resurrection to life on this earth. (John 5:28, 29) They will be given a new body that will be perfect and not die over time. (Revelation 21:4) In the meantime they are simply "asleep" in death, resting safely in God's memory. (John 11:11-14; Acts 7:60)
The exception is those who are resurrected to spirit life, but this is only a relatively select few who will serve as kings and priests.
Here are some references from official websites of Jehovah's Witnesses. I hope that these links remain pointing to the right content, but I really don't know if it may change or not:
That is from a small tract so don't expect tons of detail. It is well written, easy to read, and easy to understand.
If you are interested in learning more, I recommend requesting a free Bible study.
For that they normally use the Bible study aid What Does the Bible Really Teach? You can also view this book online and download it in EPUB or PDF format. It has several chapters and appendix sections related to your question. The first chapter is called "Is This What God Purposed?"
If you want to do some deeper research, here is some heavier stuff from the book Insight on the Scriptures:
Q. "Do Jehovah's Witnesses believe in annihilation?"
We whole heartedly believe in what is stated under divine inspiration of God's holy spirit found at 2 Thessalonians 1:6-10 and 2 Peter 3:7... and I quote:
"This takes into account that it is righteous on God’s part to repay tribulation to those who make tribulation for YOU, 7 but, to YOU who suffer tribulation, relief along with us at the revelation of the Lord Jesus from heaven with his powerful angels 8 in a flaming fire, as he brings vengeance upon those who do not know God and those who do not obey the good news about our Lord Jesus. 9 These very ones will undergo the judicial punishment of EVERLASTING DESTRUCTION from before the Lord and from the glory of his strength, 10 at the time he comes to be glorified in connection with his holy ones and to be regarded in that day with wonder in connection with all those who exercised faith, because the witness we gave met with faith among YOU." ~ 2 Thessalonians 1:6-10
"But by the same word the heavens and the earth that are now are stored up for fire and are being reserved to the day of judgment AND OF DESTRUCTION OF THE UNGODLY MEN." ~2 Peter 3:7
Consequently, contrary to popular but highly erroneous belief, we who are members of the unified worldwide Christian congregation of Jehovah's most certainly DO in fact firmly believe in Hell / Sheol / Hades / The Grave as it's stated numerous times within God's Word the Bible. However, we DON'T believe that Hell / Sheol / Hades / The Grave is a place of everlasting literal fiery torment that God Almighty sends the wicked to roast forever. In short, we believe that the Bibles teaching on what Hell / Sheol / Hades / The Grave truly is, is VERY different from what the overwhelming vast majority of the Churches of Christendom are and have been teaching people.
For example, by the use of God given reasoning skills and God's Word the Holy Bible, we find that both Adam & Eve were created PERFECT when God made them (Gen 1:31; Deuteronomy 32:4), they were flawless as far as Jehovah God was concerned regarding His standards for humans. When Adam & Eve of their own God given free-will chose to defy God and take of the forbidden fruit, they became IMPERFECT (no longer met up to God's perfect standard in human beings) and through Adam ALL of us humans inherit sin, sickness and death (Romans 3:23, 5:12).
Here's the point... Where in scripture did Jehovah God eternally send Adam to as his punishment for DELEBERATELY sinning against God?
Keep in mind also that what Adam did was not a mistake *because he knew exactly what he was doing*, he wasn't deceived / fooled into eating the forbidden fruit as his wife Eve had been (1 Timothy 2:14).
It is from and through our ancient forefather Adam that sin and death entered the world and is passed on to we who are of the human race and therefore by reason of necessity and JUSTICE, whatever Adam received as his punishment, we who are from Adam receive the exact same punishment passed on to us and our children as the inheritance of Sin:
"And Jehovah God proceeded to take the man and settle him in the garden of E′den to cultivate it and to take care of it. 16 And Jehovah God also laid this command upon the man: “From every tree of the garden you may eat to satisfaction. 17 But as for the tree of the knowledge of good and bad you must not eat from it, for in the day you eat from it YOU WILL POSITIVELY DIE.”" ~ Genesis 2:15-17
"And to Adam he said: “Because you listened to your wife’s voice and took to eating from the tree concerning which I gave you this command, ‘You must not eat from it,’ cursed is the ground on your account. In pain you will eat its produce all the days of your life. 18 And thorns and thistles it will grow for you, and you must eat the vegetation of the field. 19 In the sweat of your face you will eat bread until you return to the ground, for out of it you were taken. For dust you are and to dust you will return.”" ~ Genesis 3:17-19
Rather than a fiery place of torture / torment, Adam was sentenced back to the dust from whence he was created from, he returned back to being non-existent. That's what the Bible also shows happens to we who are his sinful descendants.
Thanks to Almighty Jehovah God through Christ there has been made a glorious way out of this otherwise hopeless condition! Again though... That's for another topic. ;-)
NOTE: I am a dedicated and baptized member of the worldwide unified Christian Congregation of Jehovah's Witnesses.