It is stated in the flashback scene of "The Man Who Would Be King" that angels do not have souls, they simply exist. When angels die, they don't go to Heaven, as can be concluded from the whole Civil War arc, because then there would just be an unlimited supply of angels. Additionally, even Purgatory is limited to creatures with a soul, which can be concluded by "The Man Who Would Be King" meaning angels don't end up there either, they would have to just "wink out" like the light on a candle being snuffed.
The same issue applies with demons, essentially. If when they were killed, they ended up back in Hell, there would simply be an unlimited supply of demons. Since they are technically just an altered human soul, though, that means there IS still a soul there. That basically just makes them supercharged ghosts, as is explained in "Weekend at Bobby's". They can additionally be killed the same way as a human spirit, by burning the bones (also "Weekend at Bobby's").
What happens to demons when they die would have to be the same thing that happens to a vengeful spirit. My personal guess is that their soul goes back to how it was before they became demons. Crowley would become Fergus McCloud again, for example.
What leads me to that conclusion is that, as shown in "Taxi Driver", when Sam went to Hell to rescue Bobby, who was previously a vengeful spirit. One could infer that they are just sent to either Heaven or Hell, but it would not make sense for them to go to any of those places as a demon; they would need to be a human soul. The other option would be that they would be sent to Purgatory. That, however, would not be explained because demons are human souls and Purgatory is "not a place for human souls," as explained in Dean's various flashbacks to Purgatory in Season 8.