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: