in this update (version 1.1.37),
inserter now can take item from both the main output and burnt result.

at long long last,we finally getting it,and thx for you and all the friends coming along contributed it and make everyone`s live much much easier.

Wow! Good news everyone!

Thanks to Factorio devs!

It was a multiple times suggested, but was not possible. (yeah, the inserter can grab fish from water, but not items from burnt result)

"Good new everyone!"
Lol, was I ninjaed here...
https://mods.factorio.com/mod/NuclearFurnace/discussion/611c4537fa7aab873f62fab9

now I found that inserter wont take burner inserter`s burnt_result,and they have no meant to fix it,so it sill got a use case,but relative minor one.
https://forums.factorio.com/viewtopic.php?f=58&t=99793