I was feeding furnaces with ore and coal. The burner inserter takes whatever's in front and that might not be a combustible item.
Stuck...

Perhaps only limit the first reach to wood, coal and solid fuel? I haven't tried but maybe rocket fuel would be relevant too?
I love this feature overall though!
There is another bug where the burner inserters don't start sometimes when something is on the far side but I can't reliably replicate it yet... That'll be for another bug report.