I sometimes dream about a problem I've been trying to solve for a day or two. The dream usually gives me an 80-90% correct solution, I just need to connect some dots.
I still remember dreaming of a kernel config issue, reading the solution on a trains timetable in a train station.
I’ll dream up a solution to a problem that’s like 50% of the way there. The other half is like “let’s use this unicorn horn to guide us to the cause of the segfault”
Some years back I've practiced lucid dreaming and managed to actively control my dreams a couple of times. While it was an exciting experience, it was definitely not as relaxing as a regular night of sleep.
So I would say, the startup's concept in general (solving work related issues while you sleep) might be doable but it would take away the night's relaxing element. So it would be more like working all night awake INSTEAD of sleeping.