Righteous Fire has the effects
Enemies Burn for 40% of your maximum Life per second as Fire Damage
Enemies Burn for 40% of your maximum Energy Shield per second as Fire Damage
indicating that the damage it inflicts is a "burning" one.
What is the difference between using a jewel with "increased fire damage" instead of one with "increased burning damage"? Which one has the largest effect on the damage I deal?
22 Answers
This page on PoE Wikipedia says
The burning damage over time is 40% of the base damage of the hit of fire damage, per second
Righteous Fire does not hit, but its burning damage per second is still increased by 40% of the base fire damage a character would deal. Therefore both modifiers have the same effect.
I tried this with Demi's RF Build using Path of Building . I removed all Jewels he used, resulting in 402887.5 DoT DPS. Adding a Jewel with 16% increased Burning Damage resulted in calculated DoT DPS of 411467.1 . Adding a 16% increased Fire Damage Jewel instead resulted also in 411467.1 DoT DPS.
3What is the difference between using a jewel with "increased fire damage" instead of one with "increased burning damage"? Which one has the largest effect on the damage I deal?
There is no difference. Use the one with the higher number, or the one which has other helpful modifiers.
All sources of Increased Damage which apply to the type of damage you are dealing are added together when calculating damage. There is no benefit to taking bonuses with multiple different sources of Increased Damage -- they all have the same effect; they just apply to different things.
Path of Exile uses the term "burning damage" as a synonym for "fire damage over time". It does not have any special meaning beyond this. In practical terms, this primarily describes damage done by igniting an enemy (which has a chance to occur when you hit them with fire damage); by the skills Righteous Fire, Scorching Ray, or Searing Bond; or by burning ground (e.g, the trails left by Flame Dash or various other skills).