When I use burning at level three, some mobs refuse to attack, they merely stand there and take damage; even creepers!
I tested this in every map, with different armors with burning, each enchant at lv.3. The mobs that are affected are below, in order of most likely in my testing (1 being most common and 10 being least common)

1. Armored Vindicators
2. Zombies
3. creepers
4. Chefs
5. Royal guards
6. Skeleton Vanguards
7. necromancers
8. Pillagers
9. Armored Zombies
10. Skeletons (Both regular and armored)