Should somehow determine whether the player is on fire and indicate that, with the estimated health loss.

Apparently the client can only know if it is on fire (doesRenderOnFire) and in what type of fire (getFireTicks).

Still can be calculated somewhat by source type and maybe it is findable after all, as wiki says

stored in a Fire tag shared by all entities

Source type is inefficient for calculation (fire = soul fire, Nether effects, fire protection enchantment...) and that Fire tag is apparently on NBT, which is server-side.