Title.

It happens sometimes, not always, and it seems like it happens to some ( random ) inserters more then the others.

Maybe there is a problem with declared Fuel value for the Omnies ?

I should add i'm using Burner Leech mod too.

Omnite can successfully fuel burner inserters. It have less fuel value than wood, but burns excellent.
Run out of fuel possible if burner inserts feeds from uncompressed faster than yellow transport belt. It has too less speed to catch up items from uncompressed speed/express belt, so cannot feed itself with fuel.

What Nucleargen said, are you using yellow or higher tier belts?

Yellow, but there is plenty of Omnite available.

Actually the problem happens more often at the beginning of the line where belt is fully compressed.

I have also noticed, that the burner inserters will not always take omnite as fuel if omnite is used as both fuel and item to be processed, and they will wait until the amount of the item to be processed is low enough to load more, and then instead od loading more Omnite to the machine, they will load it as a fuel.

Generally it seems like burner inserters are a little bit confused around machines that use Omnite as both fuel and item to be processed.