If I understand it correctly, burner inserters should be able to pick up fuel from their destination as well. So far I haven't been able to get this working. For example, I set up an assembler with a burner inserter putting output into a chest. I put coal into the chest and wait. The inserter runs out of fuel without taking any coal from the chest. Am I doing it wrong?

Work only if the destination is an assembler or a furnace with a burner inventory, and they can pick only from the burner inventory

So should this work? I was expecting the burner inserter to take coal from the furnace it is putting stone into. But it doesn't.
https://ibb.co/34B6zgp

Yes, if they put inside a furnace they can take from that furnace, but remember that the burner inserter should have the minimal energy to take items, also for steal burnable, if u have power only the furnace the inserter can take fuel from it, because is off.

Anyway if u think that the beheviour is incorrect, this is not a mod feature we use the burner leech implementation provided by the base game, so u should contact the game dev.