Destruction Warlock Tier Bonus from 4 equipped tier pieces doesn’t increase the fire damage of the trinket Erupting Spear Fragment.
The bonus from the tier set is supposed to increase all Fire damage by 1% per stack (max 8 stacks) when we have stacks of the tier bonus buff Umbrafire Embers. However, the trinket does the same amount of fire damage with or without that buff.
I have tested this several times on the target dummies to confirm it. I am not sure if it is intended or not, but seems like an oversight or poorly written set bonus description.