100% speculation territory here: Would dragons killed under the effects of dragonrend enter the dreamsleeve?

The Dreamsleeve is so depressing. There has to be some form of paradise for dragons other than the Dreamsleeve...hopefully.

You mean if there is no dragonborn nearby?

I don't think so. But I have no idea why. Dragonrend makes dragons killable, but dragons are already killable. They just keep their soul inside their body, if dragonrend can rip the soul out of a dragons body then basically using that shout would make everybody a dragonborn as it makes people capable of killing dragons permanently. There would have been no real difference between the ancient nord hero's who learned how to shout from Paarthunax and the Dragonborn who was magically gifted, and that just makes little sense.

Come to think of it the entire shout makes very little sense to me.

Oh, frak, I forgot about the dragonborn.