To put it simply, I'm inclined to say eternal torment. God won't necessarily take you out of existence, but he won't force you into his presence if you have no desire to be with him. And the lack of God leads to torment and even the inability to love.
Lysander....my experience with this teaching has been that it is rooted in the physic of man. The people who believe annihilation will not admit it but a personal aspect of annihilationism is the idea that we could not possibly be happy in heaven if we knew that some of our loved ones were suffering an eternity of torment in hell. Therefore our minds just can not accept the Bible fact of eternal torment of our loved ones.
But the TRUTH of Scriptures tell us that when we arrive in heaven, we will not have anything to complain about or be saddened by.
Rev. 21:4........
“He will wipe every tear from their eyes. There will be no more death or mourning or crying or pain, for the old order of things has passed away.”
If some of our loved ones are not in heaven, we will be in 100 percent complete agreement that they do not belong there and that they are condemned by their own refusal to accept Jesus Christ as their Savior.
John 3:16
"For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life."
It is hard to understand this, but we will not be saddened by the lack of their presence. Our focus should not be on how we can enjoy heaven without all of our loved ones there, but on how we can point our loved ones to faith in Christ so that they will be there.